Keywords: Discrete-time Scale-Shift, Hyperbolic Blaschke group, Mellin transform, Voice-Transform, Discrete Wavelet Transform, Realization

We provide a clear-cut definition of the scale-shift operator for discrete-time signals via the action of the hyperbolic Blaschke group. Practical implementation issues are discussed and given for any arbitrary scale, in the framework of very classical digital filtering. Our group theoretical standpoint leads to a purely harmonic analysis definition of the Mellin transform for discrete-time signals. Explicit analytical expressions of the atoms of the discrete-time Fourier-Mellin decomposition are provided along with a fast algorithm for their computation. The so-defined scale-shift operator also allows us to establish a mathematical equivalence in between the discrete-time wavelet coefficients of a given discrete-time signal and the corresponding Voice-transform generated by a well-chosen unitary representation of the Hyperbolic Blaschke group, in the classical Hardy space of the unit disc. This link is then, in turn, used to provide a sufficient condition to construct wavelet families from purely discrete-time signals. We also investigate the realization problem for generalized Schur and Herglotz classes of functions in Pontryagin spaces. Under a sufficient condition we detail the proof of the solution to the problem in the case of Schur by adapting the classical techniques used for the single variable problem. For the generalized Herglotz class, we proceed differently by using the notion of slice function to reduce the general problem to the single variable one. Some first topological results are obtained.

I am deeply grateful to my supervisor, Professor Mamadou Mboup, for his advice, his encouragements, his help and his rigorous supervision throughout my research over the past three years without which this work would not be completed and I wouldn't be getting a PhD degree.

All my gratitude also goes to Professor Daniel Alpay for his support through several zoom-discussions in Mathematics which have been very fruitful for me and for the realization of this thesis. I thank Mr. Hugues Mounier and Mr. Alban Goupil for accepting to participate in my second-year thesis monitoring committee. I thank Mr. Pierre Borgnat and Mrs. Paula Cerejeiras for taking their time and reading carefully this manuscript. I would also like to thanks the University of Reims and the 'Région Grand Est' for funding my doctoral researches and making my stay in France possible. I'm also very thankful to the CReSTIC Laboratory, Lab where I have been attached during these past three years. Thanks to the director, the secretary for the welcoming and the installation and all the staff for their hospitality. I would like also to take this opportunity to thanks all my friends who have brought fun to me and made my extracurricular life wonderful. In particular, I would like to give a special thanks to Mor Talla Mboup and Madiara Mbengue with whom I share many stuffs in life and have several fun and anti-stress discussions during weekends.

Last but not least, I would like to express my deep gratitude to all members of my family: my parents, my brother, all my sisters and related for their unconditional support, love and prayers all the way.

Opérateur d'échelle en temps discret et analyse harmonique associée

Résumé

La première partie de la thèse est bâtie autour d'une définition claire de l'opérateur de translation en échelle pour les signaux à temps discrets : pour un tel signal, cet opérateur est définie via l'action du groupe des transformations hyperboliques du disque (groupe de Blaschke). L'implémentation numérique de cette action est ensuite discutée et donnée, pour toute échelle arbitraire, dans le cadre du filtrage linéaire en temps discret classique.

Ce point de vue théorique nous conduit à une définition de la transformée de Mellin pour les signaux à temps discret, dans un contexte purement d'analyse harmonique. Des expressions analytiques explicites pour les atomes de la décomposition de Mellin en temps discret sont fournies avec un algorithme rapide pour leur calcul. L'opérateur de translation en échelle nous permet également d'établir une équivalence mathématique entre les coefficients d'ondelettes d'un signal à temps discret et la transformation, communément appelée "voice transform", générée par une représentation unitaire bien choisie du groupe de Blaschke dans l'espace de Hardy du disque unité (fonctions de transfert de systèmes stables et causaux). Ce lien est ensuite exploité pour construire à partir de signaux à temps discret, des familles d'ondelettes complètes dans l'espace des signaux discrets à énergie finie.

Dans une seconde partie de la thèse, nous nous intéressons également au problème de la réalisation dans le cas nD, pour des classes généralisées de fonctions de Schur et de Herglotz dans les espaces de Pontryagin. Sous une condition suffisante, nous détaillons la preuve de la solution du problème dans le cas de la classe de Schur en adaptant les techniques classiques utilisées pour ce problème en une variable. Pour la classe généralisée de Herglotz, nous procédons différemment en considérant des coupes dans les fonctions nD, permettant ainsi de ramener le problème général nD à une famille de problèmes plus simples en une variable. Quelques premiers résultats topologiques sont obtenus.

Introduction and Results Statement

Introduction Ce travail se situe dans un contexte interdisciplinaire, entre le traitement du signal et les mathématiques. Les liens entre l'analyse complexe, la théorie des opérateurs d'une part et l'analyse-réprésentation des signaux et la théorie des systèmes d'autre part ont été maintes fois exploités pour résoudre de nombreux problèmes en traitement du signal et en automatique. On peut citer par exemple le rôle de l'analyse de Schur dans la prédiction linéaire (voir par example [START_REF] Kailath | Linear systems[END_REF][START_REF] Regalia | Adaptive IIR filtering in signal processing and control[END_REF][START_REF] Dewilde | Time-varying systems and computations[END_REF][START_REF] Alpay | The Schur algorithm, reproducing kernel spaces and system theory[END_REF]), ou l'intérêt majeur des travaux célèbres de Adamian, Arov et Krein [START_REF] Vadym Adamjan | Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem[END_REF] sur l'approximation d'opérateurs de Hankel pour la synthèse de contrôleurs optimaux [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds[END_REF][START_REF] Doyle | State-space solutions to standard H 2 and H ∞ control problems[END_REF]. Comment la connection entre amibes et stabilité structurelle a permis la résolution du problème de test de la stabilité BIBO pour les systèmes rationnels multidimensionnels, pour des dimensions supérieures à 3, est un autre exemple illustré dans la publication récente [START_REF] Bossoto | Amoebas and structural stability of multidimensional systems: a test algorithm based on Monte-Carlo integration[END_REF].

Cette thèse s'inscrit dans la lignée des travaux évoqués plus haut. Elle aborde deux problèmes fondamentaux en traitement du signal par des approches d'analyse complexes. Le premier problème concerne la notion d'échelle pour des signaux à temps discret. Le second traite de la représentation de systèmes multidimensionnels dans un contexte abstrait des espaces de Pontryagin.

Le but de ce chapitre introductif est d'énoncer, sans aucunes preuves, les principaux résultats obtenus au cours de ce travail. Ainsi, dans la section 2, nous discutons de la définition de l'opérateur de translation en échelle en temps discret puis voyons comment cet opérateur conduit à une définition explicite de la transformée de Mellin dans un cadre abstrait d'analyse harmonique et qu'il permet de trouver une équivalence stricte entre les coefficients d'ondelettes discrètes et le voice transform. Dans la section 3, le problème de réalisation est abordé. Sous une condition suffisante, nous allons voir que ce problème peut être résolu pour une classe généralisée de fonctions de Schur dans un cadre abstrait des espaces de Pontryagin. Nous décrivons la suite de cette dissertation dans la section 4.

Introduction

This work is part of an interdisciplinary context, between signal processing and mathematics. The links between complex analysis, operator theory on the one hand and signal analysis-representation and systems theory on the other hand have been exploited many times to solve many problems of signal processing and automatic. One can quote for example the role of Schur analysis in linear prediction (see e.g [START_REF] Kailath | Linear systems[END_REF][START_REF] Regalia | Adaptive IIR filtering in signal processing and control[END_REF][START_REF] Dewilde | Time-varying systems and computations[END_REF][START_REF] Alpay | The Schur algorithm, reproducing kernel spaces and system theory[END_REF]) or the major interest of the famous works of Adamjan, Arov and Krein [START_REF] Vadym Adamjan | Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem[END_REF] on the approximation of Hankel operators for the synthesis of optimal controllers [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds[END_REF][START_REF] Doyle | State-space solutions to standard H 2 and H ∞ control problems[END_REF]. How the link between amoebas and structural stability allowed the resolution of the BIBO stability test problem for multidimensional rational systems is another example recently illustrated [START_REF] Bossoto | Amoebas and structural stability of multidimensional systems: a test algorithm based on Monte-Carlo integration[END_REF]. This dissertation is in the same phase with the works mentioned above. It addresses two fundamental problems in signal processing by a complex analytical approach. The first problem is concerned with the notion of scale for discrete-time signals. The second deals with the representation of multidimensional systems in an abstract context of Pontryagin spaces.

The purpose of this introductory chapter is to state, without any proof, the main results obtained during this work. So, in section 2, we discuss the definition of the discrete-time scale-shift operator then see how it leads to an explicit definition of the Mellin transform in an abstract framework of harmonic analysis and that it allows to find a strict equivalence between the discrete wavelet coefficients and the voice transform. In Section 3, the realization problem is addressed. Under a sufficient condition, this problem can be solved for a generalized class of Schur functions in an abstract framework of Pontryagin spaces. We describe the rest of this dissertation in section 4.

The discrete-time Scale-Shift Operator

Shift operators are very basic ingredients of many fundamental concepts in signal processing, such as periodicity, stationarity, linear filtering, etc. These operators also play a key role in several chapters of mathematics including functional analysis and harmonic analysis which fuel the development of linear systems theory and provide most of the signal processing tools for signal representation, signal analysis and signal synthesis. The two most common shifts, namely translation (lag operator) and scaling, are combined in the definition of the wavelet transform [START_REF] Meyer | Wavelets and operators[END_REF][START_REF] Daubechies | Ten lectures on wavelets[END_REF]. These two shift operators are linked by an anamorphosis, called the Lamperti transformation [START_REF] Lamperti | Semi-stable stochastic processes[END_REF], which corresponds to a logarithmic deformation of the time axis. This anamorphosis defines a map which allows one to express almost any concept related to time shift to its scale-shift counterpart. For instance, the Mellin transform [START_REF] Bertrand | Discrete Mellin transform for signal analysis[END_REF] appears as the image of the Fourier transform and the self-similarity property can be derived from the classical stationarity, up to some multiplicative factor [START_REF] Yazıcı | A class of second-order stationary self-similar processes for 1/f phenomena[END_REF][START_REF] Mboup | On the structure of self-similar systems: A Hilbert space approach[END_REF][START_REF] Amblard | Scale invariances and Lamperti transformations for stochastic processes[END_REF]. Unfortunately, this map applies only in the continuous-time setting. Although scaling is used in many applications, the scale-shift operator does not admit a unanimous definition in the discretetime case. In this case, scaling is generally restricted to some particular and favourable setting, as for instance the dyadic scales in discrete wavelet transform [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] or approximated, as with the principle of aggregation in the study of self-similarity [START_REF] Leland | On the self-similar nature of Ethernet traffic (extended version)[END_REF]. Yet, a clear mathematical definition exists.

Overwiew on the definition of the Scale-Shift Operator

The mathematics which lead to a rigorous and exact definition of discretetime scaling has been given in a number of places [START_REF] Alpay | Discrete-Time Multi-Scale Systems[END_REF][START_REF] Mboup | A character-automorphic Hardy spaces approach to discrete-time scaleinvariant systems[END_REF]. Its theoretical power is mainly based on its algebraic formulation, it is interpreted as the action of a specific group on the unit disc D of the complex plane C. Consider a discrete-time signal f = {f n } n⩾0 ∈ ℓ 2 and let

F (z) = n⩾ f n z n ∈ H 2 (D),
be its Z-transform. Consider the group (G β , •) where • is the composition law of functions and the set G β is given by:

G β = γ α (z) = γ 1 z + γ 2 γ2 z + γ1 , γ 1 = e iβ + αe -iβ , γ 2 = e iβ (1 -α) (1)
with |β| < π 2 fixed and α > 0. It is a subgroup of the hyperbolic Blaschke group and each element γ α can be expressed as

γ α (z) = G β • S α • G -1 β (z) (2)
where G β (z) := e iβ -z e -iβ +z and

S α : C + -→ C + z -→ αz (3)
G β is a Blaschke function and S α the scaling map in the right half plane C + .

The angle β can be fixed to zero without any restriction [START_REF] Ngom | Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals[END_REF]. The scale-shift of the signal f at scale 1 α denoted by f (γ) is defined through the following unitary operator:

T α γ : H 2 (D) -→ H 2 (D) F (z) -→ F (γ α (z)) γ2 z+ γ1 (4)
Indeed, under a basis change argument, the image of this operator can be expressed as:

T α γ F (z) = n⩾0 f (γ) n z n (5)
where

f (γ) n = l≥0 f l ϕ (γ) n,l (z), ϕ (γ) n,l (z) = ⟨Φ (γ) n (z), z l ⟩ H 2 (D) (6)
and

Φ (γ) n (z) = [γ(z)] n γ2 z + γ1 , n = 0, 1, . . . (7)
Note that {Φ (γ) n (z)} n⩾0 is an orthonormal basis of H 2 (D) for being the image under the operator T α γ of the polynomial basis {z k } k⩾0 . The signal

f (γ) = {f (γ)
n } n⩾0,γ∈G β is defined as the scale-shift of the signal f = {f n } n⩾0 at scale 1 α . Indeed, we have the following: Definition 0.1. The discrete-time scale shift operator is defined as the mapping

f = {f n } n⩾0 -→ f (γ) = {f (γ) n } n⩾0,γ∈G β . (8)
In matrix form, this reads as:

f (γ) = Φ (γ) f (9)
where the matrix

Φ (γ) = ϕ (γ) n,m n,m⩾0
and the column vectors f and f (γ) contain the samples f n , n ⩾ 0 and their scale shifted counterparts f

(γ)
n , respectively.

In a context which will be made clear in chapter 1, we replace all γ ′ s by an

α ∈ R * + for example we just write f (α) = {f (α)
n } for the scale-shifted version of the signal f = {f n } at scale 1 α .

The scalle-shift approach to the Mellin transform

The group action viewpoint of the scale-shift operator discussed above enlarges the perspective (see [START_REF] Alpay | A characterization of Schur multipliers between character-automorphic Hardy spaces[END_REF][START_REF] Alpay | Discrete-Time Multi-Scale Systems[END_REF]) and leads to an exact and purely harmonic analysis approach of the Mellin transform for discrete-time signals. Notice that this approach is in contrast with the principle of discretizing integral operators stemming from the continuous-time setting, using some adequat (geometric) grids (see e.g. [START_REF] Bertrand | Discrete Mellin transform for signal analysis[END_REF][START_REF] De | A Fast Mellin and Scale Transform[END_REF]). Indeed, we have the following [START_REF] Ngom | Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals[END_REF]:

Theorem 0.1. Let f = {f n } n⩾0 ∈ ℓ 2 be
f m (ω) = M m {f }(ω) = n⩾0 f n E m,n (ω), (10)
where the n th atom of the transform is:

E m,n (ω) = (-1) m 2π cosh(πω) P m (iω)P n (iω), n = 0, 1, 2, • • • (11)
where P n (s) is the polynomial of degree n which reads as:

P n (s) = 1 n! n k=0 n k n ℓ=1 (ℓ -k - 1 2 -s) (12)
2) -The transform is reversible and the original signal {f n } can be retrieved by:

f n = Φ (α) n (z), F (α) (z) H 2 (D) , (13)
where

F (α) (z) = m M -1 f m (s) z m
and where we use the notation M -1 for the classical inverse Mellin transform.

These polynomials P n are expressible in terms of the known Hilbert polynomials and a computational algorithm is derived which shows how easy and fast these atoms can be calculated recursively.

The scalle-shift approach to Wavelet and Voice transforms

The above discrete-time scaling approach can be used to establish a strict equivalence between wavelet coefficients and voice transform of a given discretetime signal. The theory of voice transform is introduced in [START_REF] Feichtinger | Banach spaces related to integrable group representations and their atomic decompositions. Part II[END_REF] and serves as a unifying theory for lot of atomic decomposition. See also [START_REF] Feichtinger | Banach spaces related to integrable group representations and their atomic decompositions, I[END_REF][START_REF] Pap | Properties of the voice transform of the Blaschke group and connections with atomic decomposition results in the weighted Bergman spaces[END_REF][START_REF] Pap | Hyperbolic Wavelets and Multiresolution in H 2 (T)[END_REF][START_REF] Pap | The Voice Transform on the Blaschke Group II[END_REF].

In the present context, let us first recall that the group G β , |β| > π 2 fixed, is abelian and can be unitary represented in H 2 (D) by the operator (4) in the sense that to each γ α ∈ G β one associates a unique T α γ . We make the following.

Definition 0.2. Let X(z) = n⩾0 x n z n ∈ H 2 (D) be given. For a causal discrete-time signal f = {f n } n⩾0 ∈ ℓ 2 , we define its associated Voice trans- form, generated by T α γ , with parameter X, as [V X(z) f ] : G β → C, such that [V X(z) f ](γ) = ⟨F (z), T α γ X(z)⟩ H 2 (D) = n f n x (α) n ( 14
)
where F (z) = n⩾0 f n z n .

On the other hand, it is well-known that for a signal f ∈ L 2 (R), its wavelet transform is defined by:

[W ψ f ](u, α) = 1 √ α R f (t)ψ t -u α (t)dt ( 15
)
where ψ is a mother wavelet, u ∈ R and α ∈ R * + are the time and scale shift parameters respectively.

When only sample values are available for the signal f (•) or, equivalently, one disposes a discrete-time signal f = {f n } n⩾0 ∈ ℓ 2 , then its discrete wavelet coefficients are given by:

c k (α) := n⩾0 f n h (α) n-k (16)
where h

(α) ℓ = 1 √ α ψ (α)
ℓ , {ψ ℓ } ℓ is the discrete-time version of the mother wavelet, that may be obtained, for instance, by ψ n = ψ(nT s ) where T s is a sampling time and, accordingly, ψ (α) n corresponds to the discrete-time version of ψ t α as defined in Definition 0.1 (see also [START_REF] Ngom | Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals[END_REF][START_REF] Mboup | A character-automorphic Hardy spaces approach to discrete-time scaleinvariant systems[END_REF][START_REF] Alpay | A characterization of Schur multipliers between character-automorphic Hardy spaces[END_REF]). If, in Definition 0.2, we chose as parameter X(z) = z k Ψ(z), k = 0, 1, . . . where

Ψ(z) = n⩾0 ψ n z n
then, we have the following [START_REF] Ngom | Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals[END_REF]:

Theorem 0.2. The voice transform of a discrete-time signal f = {f n } n⩾0 is equivalent to its discrete wavelet transform as given in (3.6), that is:

c k (α) = V [z k Ψ(z)] f (γ α ) (17)
This establishes a strict equivalence between wavelet coefficients and voice transform of a given discrete-time signal.

This link is exploited further to show how one can build a wavelet family from a purely discrete-time signal f = {f n } n⩾0 ∈ ℓ 2 . Indeed, consider the family f

(α) k = {f (α) k (n)} n⩾0 , k ⩾ 0 α > 0,
given by:

f (α) k (n) := f (α) n-k (18)
Let g = {g n } n⩾0 ∈ ℓ 2 be any signal which, we project in the subspace of ℓ 2 spanned by the family f

(α) k to get its coefficients ⟨g, f (α) k ⟩ ℓ 2 = n g n f (α) n-k = [V z k F (z) g](γ α ) (19)
which are expressed in terms of its voice transform parametrized from the family f

(α)

k . There exists a sufficient condition to retrieve the signal g from its coefficient [START_REF] Ball | Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables[END_REF]. Consider the following quantity

b = +∞ 0 k ⟨g, f (α) k ⟩ ℓ 2 f (α) k dα α (20)
and take its Fourier transform in discrete time. This gives b(w) = g(w)S(w) [START_REF] Benedetto | Wavelets: mathematics and applications[END_REF] where g(w) is the discrete time Fourier transform of g and

S(w) = m R m e -imw , R m = +∞ 0 r m (α) dα α (22)
where

r m (α) = n f (α) n-m f (α) n ( 23
)
If S(w) is a constant function then the following holds.

Theorem 0.3. Let f = {f n } n⩾0 ∈ ℓ 2 such that S(w) is equal to a constant C f . Then the family {f (α) k } k,α is complete in ℓ 2 that is for each g = {g n } n⩾0 ∈ ℓ 2 , we have g = 1 C f +∞ 0 k [V z k F (z) g](γ α )f (α) k dα α (24)
Moreover, there is no loss of energy

∥g∥ ℓ 2 = 1 C f +∞ 0 k V g (k, α) dα α ( 25
)
where

V g (k, α) := |[V z k F (z) g](γ α )| 2 is called scalelogram. It measures the energy of g in the time-scale neighborhood of (k, α).
This result furnishes a possibility to build complete wavelet families from discrete-time signals. Particular classes of signals, for example white noise process, satisfy this theorem. There are also profound analogies with the classical construction of wavelets which will be discussed later in this thesis.

Realization of generalized Schur and Herglotz classes

In complex analysis, the Schur class consists of those analytic functions which are defined and bounded by 1 in the unit disc or, simply, it corresponds to the unit ball of the Hardy space H ∞ (D) [START_REF] Fritzsche | Ausgewählte Arbeiten zu den Ursprüngen der Schur-Analysis: gewidmet dem großen Mathematiker Issai Schur[END_REF]. Another class close to that of Schur is the Herglotz class of analytic functions defined in the unit disc with positive real parts. These classes are intimately linked to systems theory.

Indeed, for example, given a Schur function S, there exist matrices A, B, C and D such that:

S(z) = D + zC(I -zA) -1 B ( 26
)
where the quadruple (A, B, C, D) defines the following discrete-time linear system:

     x(n + 1) = Ax(n) + Bu(n) y(n) = Cx(n) + Du(n) (27)
which has S(z) as its transfer function. From the Hilbert spaces point of view, to a Schur function S(z) is associated a reproducing kernel Hilbert space H(S) with kernel given by:

K S (w, z) = 1 -S(z)S(w) 1 -z w (28)
which is positive definite and the realization ( 26) is equivalent to the existence of a unitary operator (also called colligation):

V = A B C D : H(S) C -→ H(S) C (29)
such that its characteristic function coincides with S(z) [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF].

This way of associating a linear system with a given analytic function as its impedance is known as the realization problem.

I -S(z)S(w) = d j=1 (1 -z j wk )H j (z)H k (w) (31)
This is equivalent to the existence of a unitary colligation V as in [START_REF] Daubechies | Ten lectures on wavelets[END_REF] and

an orthogonal decomposition H = H 1 ⊕ • • • ⊕ H d of the Hilbert space H such that: S(z) = D + CZ(z)(I -AZ(z)) -1 B (32)
where

Z(z) = d k=1 z k P k with P k : H → H k is the orthogonal projection of H into H k .
The problem can be further generalized by relaxing the hypothesis [START_REF] De | A Fast Mellin and Scale Transform[END_REF] and thus letting the kernel [START_REF] Doyle | State-space solutions to standard H 2 and H ∞ control problems[END_REF] have a finite number of negative squares. This is the interest of this thesis in this part. More precisely, the generalized Schur class consists now of operator-valued funtions S : Ω(S) -→ L(F, Q) where Ω(S) is a region of the polydisc containing the origin, the coefficient spaces F, Q are Pontryagin spaces, L(F, Q) denotes the set of bounded operators defined from F to Q and such that the associated the kernel given by:

K S (w, z) = I Q -S(z)S(w) 1 -⟨z, w⟩ (33)
has a finite number of negative squares k where ⟨z, w⟩ = d j=0 z j wj . The case d = 1 is studied in [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF] with several other extensions. Using the same techniques and methods as in [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF], we give easily the extension for any d > 1.

Indeed, the following holds as long as the negative indices ind -F, ind -Q of F and Q respectively satisfy:

ind -F = ind -Q + (d -1)k (34)
Theorem 0.4. There is a unique partial coisometric colligation

V = T F G H : H(S) F -→ H(S) d Q (35)
such that, for all h ∈ H(S), f ∈ F and z = (z 1 , . . . , z d ) ∈ Ω(S)

                     (zT h)(z) = h(z) -h(0) (zF f )(z) = (S(z) -S(0))f Gh = h(0) Hf = S(0)f (36)
where

T =     T 1 . . . T d     , F =     F 1 . . .
S(z) = H + G(I H(S) -zT ) -1 zF (37)
The same problem is also considered, in this thesis, for the generalized Herglotz class of function. This class is the set of all holomorphic L(F)valued functions Φ defined in a domain Ω(Φ) ⊂ D d containing the origin and such that the kernel:

K Φ (z, w) = Φ(z) + Φ(w) 1 -⟨z, w⟩ (38)
has a finite number of negative squares. F being a Krein space.

An analogue result to the theorem above is obtained in the case of quaternionic variables [START_REF] Abu-Ghanem | Herglotz functions of several quaternionic variables[END_REF]. We work around the problem by attaching slice functions (see e.g [START_REF] Rudin | Function theory in polydiscs[END_REF] for this notion) to Φ. This allows to reduce, systematically, the general case to the one variable. The slices are already used in [START_REF] Serban | Schur coefficients in several variables[END_REF] to introduce the notion of functional Schur parameter. This technique will also allow here to leave the coefficient spaces being Krein which are generalization of the Pontryagin spaces.

Outline of the dissertation

We now give the outline of this dissertation. In chapter 1, we give the clear mathematical definition of the scale-shift operator for discrete-time signals. Then we investigate its implementation. Given a discrete-time signal

{f n } n∈N ,

Introduction

In this chapter, we deeply study the definition of the scale-shift operator for discrete-time signals. All mathematical details which are needed for a clear and rigorous definition of this operator are given. In section 2, we discuss those technical details through an analogy with the continuous time CHAPTER 1. THE SCALE-SHIFT OPERATOR AND ITS IMPLEMENTATION case which is more easy to elaborate. This analogy allows to interpret the discrete-time scale-shift operator as action of a Blaschke group on the unit disc of the complex plane. In section 3, practical implementation issues are discussed and given for any arbitrary scale, in the framework of very classical discrete-time linear filtering. In section 4, through an comprehensible example, we show how Gibbs oscillations corrupt the scale-shifted versions of finite support signals. We also provide a solution on how to mitigate the Gibbs phenomenon.

Definition of the discrete scale-shift operator

Scale-shift in continuous time

The notion of scale-shift is readily understood in the context of continuous time. Indeed, let f (t), t ∈ R + , be a given causal continuous-time signal and let F (s) be its Laplace transform (LT) given by

F (s) = R + f (t) exp (-st)dt (1.1)
The scale-shift of f (t) at scale α ∈ R + * reads as f (αt) and its LT is 1 α F s α . Scaling is thus expressed, via the operator

S α : u → S α (u) = αu (1.2)
in the same way in the two domains of continuous-time (u ∈ R + ) and frequency (u ∈ C + ). Although this is naturally elaborated, it will serve us as analogy to give an exact definition of the scale-shift operator in discrete time.

Scale-shift in discrete time

In discrete-time world, scaling a given signal f = {f n } n⩾0 does not come directly as it is in continuous-time. The key step leading to its clear mathematical definition, given in [START_REF] Mboup | A character-automorphic Hardy spaces approach to discrete-time scaleinvariant systems[END_REF][START_REF] Alpay | Discrete-Time Multi-Scale Systems[END_REF], was the observation that the operator S α is equivalent to the action of a sugbroup of the hyperbolic Blaschke group on the unit disc D of the complex plane:

G = γ(z) = γ 1 z + γ 2 γ2 z + γ1 , |γ 1 | 2 -|γ 2 | 2 = 1 (1.3)
Each element of G maps D, the unit circle T and the exterior of the closed

disc D = D ∪ T into itself respectively. For f = {f n } n⩾0 ∈ ℓ 2 , consider its Z-transform F (z) = n⩾0 f n z n , (1.4)
which converges to an element of the Hardy space of the disc H 2 (D). Now, for a given fixed β with |β| < π 2 , it is well known that the Möbius function

G β (s) = e iβ -s
e -iβ +s maps C + into D conformally [START_REF] Ford | Automorphic functions[END_REF]. Then, for any z ∈ D, there exists a unique s ∈ C + such that z = G β (s). Hence, from F (z), one considers the map defined in C + by

X(s) := F (G β (s)) (1.5)
The map X is well defined and it coincides with the LT of some causal continuous-time signal x(t), t ⩾ 0 (this is by one of the fundamental results in Fourier-Laplace analysis [START_REF] Stein | Fourier analysis: an introduction[END_REF]). In the frequency domain, the scale-shifted

signal x t α , α > 0, is represented by αX(αs) = X (S α (s)) = F (G β • S α (s)) (1.6) Replacing s by G -1 β (z) gives X S α • G -1 β (z) = F G β • S α • G -1 β (z) (1.7)
This establishes the relation between scaling a discrete-time signal, in the frequency domain, and its continuous-time counterpart. It remains now to identify exactly the discrete-time signal whose frequency representation cor-responds to (1.7) (up to normalization). This leads us to consider the family

G β = γ α (z) = G β • S α • G -1 β (z), |β| < π 2 . (1.8)
Now we have: [START_REF] Mboup | A character-automorphic Hardy spaces approach to discrete-time scaleinvariant systems[END_REF][START_REF] Alpay | Discrete-Time Multi-Scale Systems[END_REF]). Endowed with the composition law •, G β forms a group which is a subgroup of G. Each γ α ∈ G β can be written as:

Lemma 1 ([
γ α (z) = G β • S α • G -1 β (z) = γ 1 z + γ 2 γ2 z + γ1 , (1.9)
where

γ 1 = e iβ + αe -iβ and γ 2 = e iβ (1 -α). (1.10)
Conversely, every element γ(z) ∈ G is in this form for some β = β γ and some α = α γ .

Fixing β makes G β an Abelian group. In all the sequel we fix β, and we note α γ for the scale associated by (1.9) to the element γ of G β (also called the multiplier of γ, see [START_REF] Ford | Automorphic functions[END_REF]). A direct computation shows that for all γ, φ ∈ G β , we have the relation

α γ α φ = α γ•φ (1.11)
which establishes a group homomorphism between G β and the multiplicative group of R + * . From now on, the coefficients of γ α (z) in (1.10) are normalized such that

|γ 1 | 2 -|γ 2 | 2 = 1, as in (1.3): γ 1 = e iβ + αe -iβ 2 √ α cos β (1.12) γ 2 = e iβ (1 -α) 2 √ α cos β . (1.13)
Following [START_REF] Mboup | A character-automorphic Hardy spaces approach to discrete-time scaleinvariant systems[END_REF], we introduce the Hardy space unitary operator

T α γ : H 2 (D) → H 2 (D) (1.14)
defined by

T α γ F (z) := F (γ(z)) γ2 z + γ1 = n⩾0 f n Φ (γ) n (z) (1.15)
where

Φ (γ) n (z) = [γ(z)] n γ2 z + γ1 , n = 0, 1, . . . (1.16)
The transform (1.15) is just a normalization of (1.7). For each n ⩾ 0, it can be readily seen that

Φ (γ) n (z) is the image of z n under T α γ .
Notice that these operators (1.15) appear in more general setting. They are the basic tools used in the study of character automorphic functions [START_REF] Pommerenke | On Fushian groups of accessible type[END_REF][START_REF] Sodin | Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions[END_REF].

Being unitary [START_REF] Alpay | Discrete-Time Multi-Scale Systems[END_REF], T α γ preserves the scalar product and therefore the family {Φ

T α γ F (z) = n⩾0 f (γ) n z n (1.17)
where ∀n ⩾ 0,

f (γ) n = ⟨z n , T α γ F (z)⟩ H 2 (D) = m f m ϕ (γ) n,m , (1.18)
with

ϕ (γ) n,m = ⟨z m , Φ (γ) n (z)⟩ H 2 (D) . (1.19)
Therefore, we have the following:

Definition 1.1. The discrete-time scale shift operator is defined as the mapping

f = {f n } n⩾0 -→ f (γ) = {f (γ) n } n⩾0,γ∈G β . (1.20)
described in (1.15)- (1.19). In matrix form, this reads as

f (γ) = Φ (γ) f (1.21)
where the matrix

Φ (γ) = ϕ (γ) n,m n,m⩾0
and the column vectors f and f (γ) contain the samples f n , n ⩾ 0 and their scale shifted counterparts f

(γ)
n , respectively.

This operator allows one to make a direct zoom in any given discrete-time signal f = {f n } n⩾0 for any given scale 1 α γ . Now that the role of G β in this operator is clear, we invoque the homomorphism (1.11), (1.12) and (1.13) to replace all functions of γ ∈ G β by functions of α ∈ R + * . Thus, if {f n } is the discrete-time analog of f (t) then we will denote by {f

(α) n } the discrete-time analog of f (t/α), keeping in mind that α = α γ for some γ(z) ∈ G β as in Lemma 1.

Implementation of the discrete scale-shift operator

The implementation is based on a classical digital filtering process. Note that a related implementation appeared in [START_REF] Heuberger | A generalized orthonormal basis for linear dynamical systems[END_REF] in the context of system identification with Laguerre basis (see also [START_REF] Manngård | Identification of low-order models using Laguerre basis function expansions[END_REF] and [START_REF] Wahlberg | On approximation of stable linear dynamical systems using Laguerre and Kautz functions[END_REF]). We start by bringing back the functions Φ (γ) n (z) in (1. [START_REF] Ball | Canonical de Branges-Rovnyak model transferfunction realization for multivariable Schur-class functions[END_REF]) and observe that they satisfy the recurrence relation:

Φ (γ) 0 (z) = 1 γ2 z + γ1 (1.22) Φ (γ) n (z) = γ(z)Φ (γ) n-1 (z), n ⩾ 1. (1.23)
Let {δ n } denote the unit impulse sequence. Given a discrete-time signal f = {f n }, its scale-shift at a scale 1 α γ > 0, as given in (1.18), can be implemented by the following scheme:

δ n Φ 0,γ (z) ϕ n,0 γ(z) ϕ n,1 γ(z) • • • ϕ n,2 • • • ⊗ f 0 ⊗ f 1 ⊗ f 2 ⊕ ⊕ • • • • • • ⊕→ f (αγ ) n

Finite support and Gibbs phenomenon

In this final section of this chapter, we highlight the Gibbs phenomenon which comes to corrupt the scale-shifted versions of signals with finite duration.

Without loss of generality, we consider the example below to fix the ideas.

Let {f n } be the discrete-time signal defined for all n ⩾ 0 by

f n = f (nT s )
where T s = 0.01 and f (t) = 0.66 (t -1.87) 2 + 0.1 e -0.66(t-2) 2 .

(1.24)

Below, we consider the signal {f n } N n=0 and investigate the effect of the finite support N on its scale-shitfs f

(α) n N α n=0
, for various N and α. As one may readily observe, the support of the scale shifted signal is given by

N α = ⌊N α⌋
where the notation ⌊•⌋ stands for the integer part of the argument.

Smooth signal

In this first example, we set N = 600. This is large enough to consider that the whole support of f (t) is covered. The top plot in figure 1 } N n=0 defined by The error signal e

h n = f (α) n (1.25) e (α)•(β) n = f n -h (β) n (1.26) with α = 1 β = 1
( 1 √ 2 )•( √ 2) n n⩾0
is negligible, which thus shows an almost exact reversibility of the discrete-time scale-shift operation. The same conclusion can be drawn for the other error signal, e

( √ 2)•( 1 √ 2 ) n n⩾0
, although it shows some fluctuations with gradual amplitudes in the second half of its support. As explained below, these fluctuations come from the Gibbs phenomenon, induced by the fact that f N > 0 corresponds to a jump, even though the amplitude is small.

Nonsmooth signals and Gibbs oscillations

Recall that, as a transfer function, γ(z) in figure 1.1 has an infinite impulse response. Strictly speeking, the matrix Φ (γ) in (1.21) is therefore doubly infinite even though the implementation considers a limited row dimension of N α as given above, to keep only the meaningful part of the scaled signal.

The α-scale shift of a finite support signal {f n } N n=0 then reads in matrix form as

             f (α) 0 . . . f (α) N α * * . . .              = Φ (γ)                f 0 . . . f N 0 0 0 . . .                (1.27)
where the " * " are meaningless values since only the first N α + 1 components are of interest. This equation shows that the scale shift operator sees a jump in the finite support original signal, by considering its extension of infinite duration by zero padding. The jump is always present unless the last samples smoothly decay to zero, that is: f n → 0, n = N 0 , N 0 + 1, . . . , N for some range N 0 < N . As is well known, low-pass filtering a signal with jump induces a Gibbs phenomenon. Now, we claim that these Gibbs oscillations start to occur, in the scaled version of the signal, from the time index

n α = N α .
(1.28)

When α < 1 (scale contraction) we have n α > N α and thus the Gibbs phenomenon does not appear in the finite support of the scaled signal. However, it happens in the meaningless part of the scaled signal, marked by the symbol " * " in the left-hand side of (1.27).

For α > 1 however, the oscillations become visible in the whole time index interval [n α , N α ]. These conclusions are illustrated in the next example.

In this second example, we reconsider the discrete-time signal {f n } N n=0 sampled from f (t) in (1.24), with a finite duration N such that f ((N +m)T s )

is significantly different from 0, for all m = 0, 1, . . . , M , for some M > 0. In the red colored curve of the same figure, corresponding to α 2 less than 1, the Gibbs phenomenon is still present but the oscillations start after the time index N α 2 since in this experiment, we have n α 2 > N α 2 . Observe also that as α 1 = 1/α 2 , we have a coincidence of n α 1 and N α 2 . These results confirm the previous analysis on the effect of a jump on the scale operator.

Note also that the analysis already suggests a solution to make the scale shift operator safe from the Gibbs phenomenon.

Mitigating the Gibbs phenomenon

To begin, recall that for α < 1, the Gibbs oscillations are unseen because they happen after the time N α , that is in the meaningless part of the scaled signal (see equation (1.27)). Now, observe that one can get back to the same situation with α > 1 by pushing (artificially) the jump of the signal far enough. More precisely, consider the computation of the α-scale shift of the signal {f n } N n=0 (with jump at the end, as explained before), with α > 1. First we define the signal

{ f n } N n=0 of duration N = N + K α such that f n =      f n , n = 0, 1, • • • , N f N n = N + 1, • • • N = N + K α (1.29)
where K α is some constant depending on α. According to eq.(1.27) and setting N α = ⌊ N α⌋, the α-scale shift of { f n } N n=0 then reads as:

                    f (α) 0 . . . f (α) N α f (α) N α+1 . . . f (α) N α * . . .                     = Φ (γ)                    f 0 . . . f N f N . . . f N 0 . . .                             K α lines (1.30)
We thus get back to the setting in eq.(1.27) where the original jump, at time index N , is now carried over to the time index N = N +K α . Accordingly, the Gibbs oscillations start to occur at the time index

n α = N α .
N α = ⌊N α⌋ ⩽ n α = N + K α α .
Now this is achieved when K α ⩾ N (α 2 -1) . This is confirmed by the results in figure 1.5, where the scaled versions {f

(α)
n } N α n=0 of the finite support signal {f n } N n=0 (black curve) are free from any Gibbs oscillations for all α, be it lower or greater than 1. The scaled version is obtained from the filtering scheme in figure 1.1 equation (1.27), for α < 1 (red curve in the plot). For α > 1, it is defined by L'approche que nous proposons est différente de celle basée sur des discrétisations d'intégrales issues du cadre en temps continu avec l'utilisation de certaines grilles adéquates (voir e.g. [START_REF] Bertrand | Discrete Mellin transform for signal analysis[END_REF][START_REF] De | A Fast Mellin and Scale Transform[END_REF]). Dans la section 2, nous révisons brièvement les éléments l'analyse harmonique qui sont à la base de la transformée de Fourier abstraite. Noter que ce domaine est standard et peut être lu dans de nombreux manuels tels que [START_REF] Rudin | Fourier analysis on groups[END_REF][START_REF] Goldberg | Fourier transforms[END_REF] et [START_REF] Hewitt | Abstract Harmonic Analysis[END_REF]. Dans la section 3, nous montrons à quel point la définition de la transformée de Mellin pour les signaux à temps discret s'inscrit facilement et naturellement dans ce cadre général. Dans la section 4, nous présentons un algorithme rapide pour le calcul des atomes de cette décomposition.

f (α) n = f (α) n , n = 0, • • • , N α (blue curve).

Introduction

Introduction

In this chapter, by use of the discrete scale-shift operator, we provide an exact and purely harmonic analysis definition of the Mellin transform for discrete-time signals. This transform is a well-known tool in mathematics for its wonders in solving functional equations, in analyzing algorithms, in combinatorial analysis, etc. The usefulness of this transform stems from its basic properties, which are very easy to establish, but also from its remarkable asymptotic properties. The proposed approach is in contrast with the principle of discretizing integral operators stemming from the continuoustime setting, using some adequat (geometric) grids (see e.g. [START_REF] Bertrand | Discrete Mellin transform for signal analysis[END_REF][START_REF] De | A Fast Mellin and Scale Transform[END_REF]). In section 2, we make a brief review of the basic elements of harmonic analysis that underlie the abstract Fourier transform. This subject is standard and is exposed in a number of classical (re-edited) textbooks such as e.g. [START_REF] Rudin | Fourier analysis on groups[END_REF][START_REF] Goldberg | Fourier transforms[END_REF] and [START_REF] Hewitt | Abstract Harmonic Analysis[END_REF]. In section 3, we show how easy and natural the definition of the Mellin transform for discrete-time signals falls within this general framework.

In section 4, we provide a fast algorithm to compute the atoms.

Review of Abstract Harmonic Analysis

Abstract harmonic analysis is built on locally compact groups. These groups play key roles in Mathematics. They admit a special measures called Haar measures which allow to define on them integrals of measurable functions.

From now on, (G , •) denotes a locally compact Abelian group, with Haar measure µ, and neutral element ι.

i) σ(ι) = 1 (2.1) ii) σ(a • b) = σ(a)σ(b), ∀a, b ∈ G (2.2)
is called a character.

The set of all continuous characters forms a group G called the dual group of G . By the Pontryagin duality, G is also a locally compact Abelian group and we denote by μ its asociated Haar measure. Now, for f : G → C in L 1 (G , dµ), we have Definition 2.2. The Fourier transform of f F and its inverse F -1 are defined respectively by:

f (σ) = F{f }(σ) = G f (a)σ(a)dµ(a), σ ∈ G (2.3) f (a) = F -1 { f }(a) = G f (σ)σ(a)dμ(σ), a ∈ G .
(2.4)

Group action

Consider the action of G on a non-empty set E. This can be described via a

morphism η : G × E → E such that • η(ι, t) = t, for all t ∈ E • η(a, η(b, t)) = η(a • b, t), ∀a, b ∈ G and ∀t ∈ E.
To every f : E → C, one can thus associate a function

h : G × E → C defined for all (a, t) ∈ G × E by h(a, t) = f • η(a,
h t (a) = h(a, t) = f (η(a, t)) = f (η(a, η(b t , t 0 ))) = f (η(a • b t , t 0 )) = h t 0 (a • b t ),
showing that the two variables merge into one and the function h(•, •) reduces to a function of a single variable. The Fourier transform of h t (•) becomes:

h t (σ) = G h t (a)σ(a)dµ(a) = G h t 0 (a • b t )σ(a)dµ(a) = G h t 0 (a)σ(a • b -1 t )dµ(a • b -1 t ) = σ(b t ) h t 0 (σ),
where the last equality is obtained using (2.1)-(2.2) and the translationinvariance property of the Haar measure.

An example of such simple situation is the case when G is the additive group of real (resp. integer) numbers, and E = G : The Haar measure is the Lebesgue measure (resp. counting measure) and the characters are the exponential functions σ ω (a) = e -iωa . This corresponds to the classical Fourier transform. Another common example is obtained with the multiplicative group of positive real numbers, with again, E = G : The Haar measure is dµ(a) = da a , the characters are the functions σ ω (a) = a -iω , ω ∈ R and we recover the classical Mellin transform which is then given, for an

f ∈ L 1 (R * + , da a ), as M{f }(w) = +∞ 0 f (a)a iw da a (2.6)
Notice that this transform is reversible and we can retrieve f from its coefficients (2.6) as

f (a) = 1 2πi c+i∞ c-i∞ M{f }(w)a -iw dw (2.7)
where c belongs to the fundamental domain of convergence of the transform (2.6).

We provide exact definition of (2.6) in case where f is now a discrete-time signal.

The Mellin transform for discrete-time signals

As observed already, scale translation in discrete-time is expressed via the action of the group G β (β being fixed as before). We recall that by Lemma 1,

(1.11) and Definition 1.1, this action is equivalent to that of the multiplicative group of positive reals, with E = N. The associated morphism η(α, n) is defined such that for a given discrete-time signal f = {f n } n⩾0 with its Ztransform F (z) = f n z n , we have:

f η(α,n) := f (α) n = ⟨z n , T α F (z)⟩ H 2 (D) . (2.8)
Clearly, this action is not transitive. Therefore, the two variables α ∈ R + * and n ∈ N cannot be merged. For any fixed pair (n, m) of positive integers

with m ̸ = n, f (α)
n and f

f m (ω) = M m {f }(ω) = n⩾0 f n E m,n (ω), (2.9)
where the n th atom of the transform is

E m,n (ω) = (-1) m 2π cosh(πω) P m (iω)P n (iω), n = 0, 1, 2, • • • (2.10)
where P n (s) is the polynomial of degree n which reads as

P n (s) = 1 n! n k=0 n k n ℓ=1 (ℓ -k - 1 2 -s) (2.11)
2) -The transform is reversible and the original signal {f n } can be retrieved by

f n = Φ (α) n (z), F (α) (z) H 2 (D) , (2.12)
where

F (α) (z) = m M -1 f m (s) z m
and where we use the notation M -1 for the classical inverse Mellin transform.

Proof. The proof is given in appendix A.

In the sequel, we define the polynomials

p k,n (s) = n ℓ=1 (ℓ -k -s - 1 2 ), n ∈ N, 0 ⩽ k ⩽ n (2.13)
so that the polynomial in (2.11) reads as

P n (s) = 1 n! n k=0 n k p k,n (s).
In the next section we derive from these polynomials a fast algorithm for computing the atoms (2.10).

Computation Algorithm

The following lemma shows some relations on these polynomials (2.13) that are useful for the computation of the atoms E m,n (s).

Lemma 2. The polynomials p k,n satisfy the following recurrence relations

p n-k,n (s) = (-1) n p k,n (-s) (2.14) p k,n (s) = 2(n -k) -1 2 -s p k,n-1 (s), ∀n ⩾ 1 (2.15) p k,n (s) = -2k+1 2 -s 2(n-k)+1 2 -s p k-1,n (s), 1 ⩽ k ⩽ n (2.16)
Proof. See appendix A.

Now we have:

Lemma 3. On the imaginary axis, s = iω, ω ∈ R, the polynomial P n (s) is either real valued or purely imaginary and we have:

1. For n even, n = 2q,

P 2q (iω) = 2 (2q)! q-1 k=0 2q k ℜ {p k,2q (iω)} + 1 (q!) 2 p q,2q (iω)
(2.17) and p q,2q (iω) is real.

2. For n odd, n = 2q + 1,

P 2q+1 (iω) = 2i (2q + 1)! q k=0 2q + 1 k ℑ {p k,2q+1 (iω)} (2.18)
Proof. See Appendix A.

These lemmas show how one can compute the polynomials P n (s) for all n, using simple recurrences initialized from p 0,0 (s) = 1. Indeed, for each given n > 0, the polynomials p k,n (s), k = 1, • • • , n can be easily obtained recursively from p 0,n (s) by following the recurrence relation (2.16). Now, the polynomial p 0,n (s), in turn, easily derives from p 0,0 (s) by a recursive application of the relation (2.15). Finally, the polynomials p k,n (s), and hence P n (s), can be computed using the following diagram where the vertical arrow with the circle mark in the middle represents the relation (2.16) and the one with the square mark represents the relation (2.15).

p 0,0 | ⇝ P 0 . . . p 0,2(q-1) • • • p q-1,2(q-1) | ⇝ P 2(q-1) p 0,2q-1 • • • p q-1,2q-1 | ⇝ P 2q-1 p 0,2q • • • p q-1,2q p q,2q | ⇝ P 2q p 0,2q+1 • • • p q-1,2q+1 p q,2q+1 | ⇝ P 2q+1 . . . . . . . . .
Based on Lemma 3 and (2.10), it holds that E m,n (ω) is real when n + m is even and purely imaginary otherwise. The supports and the shapes of the different atoms vary markedly with m and n. This is illustrated in figure 2.1 below, where the atoms or their imaginary part were appropriate, are represented for some values of n and m.

Introduction

In this chapter, by use of the discete scale-shift operator discussed in chapter 1, we establish a strict equivalence between two popular signal transforms:

the discrete wavelet transform and the Voice transform for the Blaschke group. This equivalence even if it was studied and approached in a number of places, it has not been yet proved mathematically. We prove it here by invoking just the subgroup (1.8) of the whole Blaschke group and fix the angle β. This link prompted us to study how one can construct a wavelet family from a discrete-time signal using the scale-shift operator. We provide a sufficient condition to make that possible. In section 2, we review the basic notions in wavelet theory and the voice transform which will be needed in the sequel. In section 3, we prove the equivalence between the voice transform and the wavelet coefficients of a discrete-time signal. In section 4, we provide a sufficient condition to construct a wavelet family from a purely discrete-time signal.

Preliminaries

Brief review of wavelet theory

In wavelet theory (see e.g [START_REF] Meyer | Wavelets and operators[END_REF][START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF][START_REF] Hernández | A first course on wavelets[END_REF][START_REF] Benedetto | Wavelets: mathematics and applications[END_REF]), it is well known that one starts from a zero average function ψ ∈ L 2 (R), called mother wavelet, and then build a wavelet family

ψ u,α (t) = 1 √ α ψ t -u α (3.1)
where u ∈ R, α ∈ R * + are the time and scale shift parameters respectively. The wavelet coefficients of a given signal f ∈ L 2 (R) are defined as

[W ψ f ](u, α) := ⟨f, ψ u,α ⟩ L 2 (R) = R f (t)ψ u,α (t)dt (3.2)
Provided that the following admissible condition, expressed in terms of the Fourier transform of the mother wavelet,

C ψ = ∞ 0 | ψ(w)| 2 w dw < ∞, (3.3)
is satisfied, the signal f can be recovered from its wavelet coefficients (3.2) as

f (t) = 1 C ψ +∞ 0 +∞ -∞ [W ψ f ](u, α)ψ u,α (t) dudα α 2 (3.4)
Moreover, there is no loss of energy

R |f (t)| 2 dt = 1 C ψ +∞ 0 +∞ -∞ |[W ψ f ](u, α)| 2 dudα α 2 (3.5)
Note that, in general, the family {ψ u,α (t)} u,α is very redundant so that it can't form a basis for L 2 (R).

The discrete wavelet transform is introduced according to the choice of the discreteness of time and scale parameters u and α respectively. The choice of dyadic scales is very special for fast computational issues. All constructed orthogonal wavelet bases (see e.g [START_REF] Meyer | Construction de bases orthonormées d'ondelettes[END_REF][START_REF] Daubechies | Orthonormal bases of compactly supported wavelets[END_REF][START_REF] Gilles | Ondelettes et bases hilbertiennes[END_REF]) are established under that dyadic scales choice.

In our context, if the signal f (•) is only known at sample values or, equivalently, one disposes a discrete-time signal f = {f n } n⩾0 ∈ ℓ 2 , then its discrete wavelet coefficients are defined by:

c k (α) := n⩾0 f n h (α) n-k (3.6)
where

h (α) ℓ = 1 √ α ψ (α) ℓ (3.7)
{ψ ℓ } is the discrete-time version of the mother wavelet, that may be obtained, for instance, by

ψ n = ψ(nT s )
where T s is a sampling time and, accordingly, ψ

n corresponds to the discretetime version of ψ t α as defined in Definition 1.1 (see also [START_REF] Ngom | Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals[END_REF], [START_REF] Mboup | A character-automorphic Hardy spaces approach to discrete-time scaleinvariant systems[END_REF], [START_REF] Alpay | A characterization of Schur multipliers between character-automorphic Hardy spaces[END_REF]).

The Voice transform

The theory of Voice transform was originally introduced in [START_REF] Hans | Banach spaces related to integrable group representations and their atomic decompositions. Part II[END_REF] and serves as a unifying theory for lot of atomic representations (see also [START_REF] Pap | Properties of the voice transform of the Blaschke group and connections with atomic decomposition results in the weighted Bergman spaces[END_REF][START_REF] Pap | The Voice Transform on the Blaschke Group II[END_REF][START_REF] Soumelidis | Applying hyperbolic wavelet constructions in the identification of signals and systems[END_REF]). Generally speaking, one starts by representing a given locally compact topological group G in a given Hilbert space H by unitary operators

U g : H → H, g ∈ G , defined in H.
More precisely, the voice transform of an element f ∈ H generated by U g , g ∈ G , and with respect to some parameter h ∈ H, is defined as

[V h f ](g) = ⟨f, U g h⟩ H (3.8)
where ⟨•, •⟩ H is the inner product in the Hilbert space H.

Related to the present context, let us consider the subgroup G β defined in (1.8), with an angle β ∈ (-π/2, π/2) fixed as before. Recall that the operators

T α γ : H 2 (D) → H 2 (D)
defined in (1.15) are unitary representations of G β in the Hilbert space of the disc, H 2 (D).

Therefore, we can define the Voice transform with respect to G β , for discrete-time signals:

Definition 3.1. Let X(z) = n⩾0 x n z n ∈ H 2 (D) be given. For a causal discrete-time signal f = {f n } n⩾0 ∈ ℓ 2
, we define its associated Voice transform, generated by T α γ , with parameter X, as:

[V X(z) f ] : G β → C, such that [V X(z) f ](γ) = ⟨F (z), T α γ X(z)⟩ H 2 (D) = n f n x (α γ ) n (3.9)
where F (z) = n⩾0 f n z n .

Voice transform to Wavelets coefficients

We are now ready to state and prove the mathematical equivalence between the Voice transform and the wavelet coefficients of a given discrete-time signal. We mention that the relation in between these two transforms has been studied in [START_REF] Soumelidis | Applying hyperbolic wavelet constructions in the identification of signals and systems[END_REF] without touching an explicit and strict equivalence.

c k (α) = V [z k Ψ(z)] f (γ α ) (3.10) Proof. Let F (z) = n⩾0 f n z n and Ψ(z) = n ψ n z n
be the Z-transforms of the sequences {f n } n⩾0 and {ψ n } n⩾0 respectively.

The Z-transform of the discrete-time scale shifted signal ψ (α) n then readily reads as

Ψ (α) (z) = n ψ (α) n z n = T α Ψ(z).
Likewise, for any fixed k ∈ N, we will use the notation:

Ψ (α) k (z) = n ψ (α) n-k z n = T α [z k Ψ(z)]
to represent the discrete-time scale shift of {ψ n-k } n⩾0 . The proof follows upon noting that the expression (3.6) also reads in a form of a scalar product

c k (α) = n f n ψ (α) n-k (3.11) = ⟨F (z), Ψ (α) k (z)⟩ H 2 (D) (3.12) = ⟨F (z), T α z k Ψ(z)⟩ H 2 (D) = V [z k Ψ(z)] f (γ α ), (3.13)
where γ α (z) is the element of G β associated with the scale α as in (1.8).

Discrete-time Wavelet family

The above established equivalence between discrete wavelet coefficients and the voice transform suggest clearly another close study which investigates on how one obtain a complete wavelet family by going from a purely discretetime signal whose corresponding coefficients will be expressible as in (3.10).

Consider a discrete-time signal f = {f n } n⩾0 ∈ ℓ 2 . We build the family

f (α) k = {f k,α (n)} n⩾0 given by f k,α (n) := f (α) n-k (3.14)
The purpose is to establish sufficient conditions on the generating signal f such that the family {f k,α } k,α is complete or even a basis in ℓ 2 .

Let g = {g n } n⩾0 ∈ ℓ 2 be any signal which, we project in the subspace of ℓ 2 defined by the family f

(α) k to get its coefficients ⟨g, f (α) k ⟩ ℓ 2 = n g n f (α) n-k = [V z k F (z) g](γ α ) (3.15)
which are expressed in terms of its voice transform parametrized from the family f

(α)

k . There exists a sufficient condition to retrieve the signal g from its coefficient (3.15). Consider the following quantity

b = +∞ 0 k ⟨g, f (α) k ⟩ ℓ 2 f (α) k dα α (3.16)
b is a vector with components given by

b n = +∞ 0 k ⟨g, f k,α ⟩ ℓ 2 f (α) n-k dα α (3.17)
It turns out that the expression in parenthesis can be written as a discretetime convolution

b n = +∞ 0 g ⋆ f (α) ⋆ f (α) n dα α (3.18)
where

f (α) = { f (α) n } with f (α) n =      f (α) n if n ≥ 0, f (α)
-n else..

(3.19)
Taking the Fourier transform in n in (3.18) gives

b(w) = g(w) +∞ 0 f (α) (w) f (α) (w) dα α (3.20)
where

f (α) (w) = n f (α) n e -inw (3.21) Therefore, b(w) = g(w) +∞ 0 n l f (α) l f (α) n e -i(n+l)w dα α (3.22) = g(w) +∞ 0 n m f (α) m-n f (α) n e -imw dα α (3.23) Assuming that m ≤ n, (3.23) becomes b(w) = g(w) +∞ 0 n m f (α) n-m f (α) n e -imw dα α (3.24) = g(w)S(w) (3.25)
where

S(w) = m R m e -imw , R m = +∞ 0 r m (α) dα α (3.26)
and

r m (α) = n f (α) n-m f (α) n (3.27)
If S(w) is a constant function then the following result holds.

Theorem 3.2. Let f = {f n } n⩾0 ∈ ℓ 2 such that S(w) is equal to a constant C f . Then the family {f (α) k } k,α is complete in ℓ 2 that is for each g = {g n } n⩾0 ∈ ℓ 2 , we have g = 1 C f +∞ 0 k [V z k F (z) g](γ α )f (α) k dα α (3.28)
Moreover, there is no loss of energy

∥g∥ 2 ℓ 2 = 1 C f +∞ 0 k V g (k, α) dα α (3.29)
where

V g (k, α) := |[V z k F (z) g](γ α )| 2 is called scalelogram.
It measures the energy of g in the time-scale neighborhood of (k, α).

Proof. The reconstruction formula (3.28) is a consequence of (3.25).

To prove (3.29), we have:

⟨g, f k,α ⟩ ℓ 2 = n g n f (α) n-k = +∞ n=k g n f (α) k-n (3.30) = (g ⋆ f (α) ) k . (3.31)
Taking the Fourier transform in k in the last equality above, we have

⟨g, f k,α ⟩ ℓ 2 (w) = ĝ(w) f (α) (w) (3.32)
By the Plancherel identity, we have:

k |⟨g, f k,α ⟩ ℓ 2 | 2 = ∥ ⟨g, f k,α ⟩∥ 2 ℓ 2 = ∥ĝ∥ 2 k |f (α) k | 2 (3.33)
Therefore integrating gives

+∞ 0 k |⟨g, f k,α ⟩ ℓ 2 | 2 dα α = ∥ĝ∥ 2 ℓ 2 +∞ 0 k |f (α) k | 2 dα α =C f (3.34) = C f ∥g∥ 2 ℓ 2 (3.35)
Note also that we put all constants intervening in the use of Plancherel identity in C f .

There exists two classes of signals for which the unique hypothesis in the theorem above which is S(w) = C f is satisfied:

1) If the signal f = {f n } n≥0 is a decorrelated finite variance stochastic process then its scle-shift f (α) = {f (α)
n } n is also of the same class and by the decorrelation r m (α) = 0, ∀m ̸ = 0. Therefore

S(w) = R 0 = +∞ 0 k [f (α) k ] 2 dα α = C f < ∞ (3.36)
∼ variance on all scales (3.37) Expressions (3.28), (3.29) hold and moreover, it can be seen from (3.27) that the family {f k,α } k,α is orthogonal and is therefore a basis for ℓ 2 .

As examples, consider a white noise process or a centered iid Gaussian random process f = {f n } n⩾0 . Decorrelation is ensured in both examples and so f belongs to the class of signals considered in this case.

-Discrete-time scale-shift meets Compressive sensing

The Discrete-time scale-shift approach to wavelets using white noise meets trivially a solution to the new theory of Compressive Sensing (CS).

Roughly, CS consists of finding possibilities to reconstruct an unknown signal from a small number of measurements taken through a so-called sensing matrix. This matrix can be built here from the family {f k,α } k,α by defining the n th column to be

f k,α (n) = [f (α) n-k , k, α], n = 0, 1, . . . M .
It is an orthogonal Gaussian matrix and therefore fulfills the incoherence and restricted isometric properties [START_REF] Baraniuk | Compressive sensing [lecture notes[END_REF] needed to be able to recover some K-sparse and compressible finite N-length signals with N >> M , as long as M ⩾ cK log (N/K), c a small constant. A large exposition on compressive sensing can be found in among others [START_REF] Candes | Near-optimal signal recovery from random projections: Universal encoding strategies?[END_REF][START_REF] Candes | Sparsity and incoherence in compressive sampling[END_REF][START_REF] Candes | Stable signal recovery from incomplete and inaccurate measurements[END_REF][START_REF] Donoho | Compressed sensing[END_REF] etc.

-Discrete scale-shift meets vanishing moments wavelets

One of the marvels of classical wavelet theory is its ability to detect local irregularities in signals. Irregularity detection is connected to the notion of vanishing moments. Indeed the following theorem gives a characterization [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF].

Theorem 3.3. A wavelet ψ with a fast decay has n vanishing moments if and only if there exists a function θ with a fast decay such that

ψ(t) = (-1) n d n θ(t) dt n (3.38)
Therefore, the wavelet coefficients (3.2) become

[W ψ f ](u, α) = α n d n dt n (f ⋆ θα )(u) (3.39)
where

θα (t) = 1 √ α θ -t α .
By the decay property of θ, the following can be shown if

f is n times continuously differentiable around u [52] lim α→0 [W ψ f ](u, α) α n+ 1 2 = Kf (n) (u) (3.40)
This gives a link between the differentiability of f around u and the decaying of its wavelet coefficients at fines scales. This result works well in the discrete-time scale-shift approach discussed in this thesis if θ is a Gaussian. Indeed, consider the signal in the figure below and compute its wavelet coefficients (3.15) using a 6th order derivative of a Gaussian as wavelet (which is equivalent to considering a wavelet with 6 vanishing moments according to the theorem above). The break creates high amplitude coefficients (yellow color in image below) which tend towards the break location at fine scales. To treat rigorously the discrete-time case, we need to introduce some concepts in discrete time. One says that a signal

f = {f n } n⩾0 ∈ ℓ 2 has ℓ vanishing moments if n⩾0 n(n -1) • • • (n -k)f n = 0, ∀0 ⩽ k < ℓ (3.41)
This definition will allows us to use the Z-transform of f to prove the following characterization theorem.

Theorem 3.4. A discrete-time signal f = {f n } n⩾0 ∈ ℓ 2 satisfying
n f n = 0 has ℓ vanishing moments if and only if there exists {θ n } n⩾0 ∈ ℓ 2 such that

f n = (-1) ℓ ℓ m=0 ℓ m (-1) m θ n-m , ∀n ⩾ 0 (3.42)
Proof. If f has ℓ vanishing moments then ∀m < ℓ, its Z-transform satisfies

F (m) (1) = n⩾0 n! (n -m)! f n = 0.
1 is then a root with multiplicity ℓ of F and so we can factorize

F (z) = (z -1) ℓ Θ(z), with Θ(z) ∈ H 2 (D) and Θ(1) ̸ = 0. It exists therefore {θ n } n⩾0 ∈ ℓ 2 such that Θ(z) = n θ n z n . So we have F (z) = (z -1) ℓ Θ(z) (3.43) = k ℓ m=0 ℓ m (-1) ℓ-m θ k-m z k (3.44)
The signal f = {f n } n⩾0 is therefore given as

f n = ⟨F (z), z n ⟩ H 2 (D) = ℓ m=0 ℓ m (-1) ℓ-m θ n-m , ∀n ⩾ 0
which gives the difference equation (3.42) and which is also the discretetime counterpart of the differential operation (3.38).

The only if part follows directly from (3.42) by taking the Z-transform and proceed.

As a consequence, the wavelet coefficients (3.15) of a given signal g = {g n } n⩾0 ∈ ℓ 2 can be written as

[V z k F (z) g](γ α ) = ℓ m=0 ℓ m (-1) ℓ-m (g ⋆ θ(α) ) k+m
where θ(α) = { θ(α) n } n is defined as in (3.19). Using the definition of the scale-shift given in Definition 1.20, we prove

lim α→0 + θ(α) n = lim α→0 + k θk ϕ (α) n,k = k (-1) k θk (3.45)
If moreover the process {θ n } n is exponentially decaying then the following holds in the weak sense

lim α→0 + θ(α) n = Cδ, C constant (3.46)
Therefore, 2) If the signal f = {f n } n≥0 is a discrete-time stochastic process not decorrelated but satisfies R m = 0, ∀m ̸ = 0 then the hypothesis in theorem (3.2) is satisfied and expressions (3.28) and (3.29) hold. However, note that in this case, orthogonality for the family {f k,α } k,α is not verified since (3.27) may not vanish.

lim α→0 + [V z k F (z) g](γ α ) = C ℓ m=0 ℓ m ( -
Remark. In both cases we have the sufficient condition

R m = +∞ 0 r m (α) dα α = 0, ∀m ̸ = 0 (3.47)
It can be interpreted as the discrete version of the wavelet admissible condition (3.3). Indeed, consider the function

α → r m (α)
Its Fourier-Mellin transform, denoted by R m , reads as

R m (ω) = +∞ 0 r m (α)α iω dα α (3.48)
Hence, condition (3.47) which is sufficient to generate a wavelet family complete in ℓ 2 is actually equivalent to R m (0) = 0, ∀m ̸ = 0. It is also possible to see that the wavelet admissibility condition (3.3) is satisfied once the Fourier transform of the wavelet mother vanishes at the origin ψ(0) = 0 and that it admits a certain regularity [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. variable problem [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF]. In section 4, the Herglotz case is discussed. We first give the realization in the one-variable case whose the proof techniques can be deduced in many places (see e.g [START_REF] Abu-Ghanem | Herglotz functions of several quaternionic variables[END_REF][START_REF] Alpay | Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions[END_REF][START_REF] Ball | Schur-Agler and Herglotz-Agler classes of functions: positive-kernel decompositions and transfer-function realizations[END_REF]). For the general case, we work around the problem by attaching slice functions to each Herglotz function Φ and then reduce, systematically, the problem to the one variable case. This slice mapping turns out to be continuous and a density result between the involved reproducing kernel Pontryagin spaces is given. Notice that the slices are already used in [START_REF] Serban | Schur coefficients in several variables[END_REF] to introduce the notion of functional Schur parameters. They also appear in many other places regarding the same issue as here but in different contexts (see e.g [START_REF] Alpay | Slice hyperholomorphic Schur analysis[END_REF][START_REF] Alpay | Schur functions and their realizations in the slice hyperholomorphic setting[END_REF][START_REF] Alpay | Realizations of slice hyperholomorphic generalized contractive and positive functions[END_REF]).

Introduction

Preliminaries

Back to the classic realization problem

In complex analysis, the theory of analytic functions is a very rich field that plays key roles in several mathematical chapters but also in systems engineering and control. In system theory, for instance, they are at the origin of the famous realization problem which consists, for a given analytical function defined over a domain of the complex plane, in finding a linear system of which this given function is its impedance. It is known that Schur's operatorvalued functions provide solution to this problem. This fueled the study of the Schur's algorithm [START_REF] Schur | Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind[END_REF][START_REF] Alpay | The Schur algorithm and reproducing kernel Hilbert spaces in the ball[END_REF] and interpolation problems [START_REF] Nevanlinna | Uber beschrankte Funktionen, die in gegeben Punkten vorgeschrieben Werte annehmen[END_REF].

By studying this problem in the multivariable framework, J. Agler [START_REF] Agler | On the representation of certain holomorphic functions defined on a polydisc[END_REF]

I -S(z)S(w) = d j=1 (1 -z j wj )H j (z)H j (w) (4.2)
This in turn is equivalent to the existence of a unitary colligation:

V = A B C D : H (
S(z) = D + CZ(z)(I -AZ(z)) -1 B (4.4)
where Z(z) = d k=1 z k P k with P k : H(S) → H k is the orthogonal projection of H(S) into H k . This class of function was a pioneer in several works among which we can cite those dealing with interpolation theory (see e.g [START_REF] Ball | Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables[END_REF][START_REF] Alpay | On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains[END_REF][START_REF] Ball | Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function in Cn[END_REF]) as well as those relating to the study of stability for multidimensional linear sytems via the notion of stable polynomials (see e.g [START_REF] Malakorn | Feedback Control for Multidimensional Linear Systems and Interpolation Problems for Multivariable Holomorphic Functions[END_REF][START_REF] Knese | Stable symmetric polynomials and the Schur-Agler class[END_REF][START_REF] Borcea | Multivariate Pólya-Schur classification problems in the Weyl algebra[END_REF][START_REF] Grinshpan | Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials[END_REF]).

The generalization of the realization problem leading to the Schur-Agler class can be furthermore extended by relaxing the condition (4.1). This generalization in which we are interested in this chapter of this thesis is formalized in the context of Pontryagin spaces or more generally Krein spaces.

Generalities on Kreins spaces

Krein spaces are generalization of Hilbert spaces whose positive definiteness of scalar products is relaxed but completeness and nondegeneracy are kept [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF].

By a Krein space, we mean a vector space K over the complex numbers C such that:

-it is endowed with a mapping

⟨•, •⟩ : K × K → C (4.5)
which is linear in the first argument and symmetric,

-it has a fundamental decomposition i.e K = K + ⊕ K -where K ± are subspaces of K and K + is a Hilbert space, K -is an antispace of a

Hilbert space [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF].

The dimensions of the spaces K ± which can be natural integers or infinite are called positive and negative index of K respectively and denoted by ind ± (K).

They are independent to the choice of the fundamental decomposition. Pontryagin spaces are those Krein spaces which have finite negative indices. They differ from Hilbert spaces just in a finite dimensional nondegenerated part.

The notation L(F, Q) stands for the space of continuous linear operators on a Krein space F into a second Krein space Q and when F = Q we just write L(F). An operator L ∈ L(F, Q) is said to be coisometric if its adjoint

L * ∈ L(Q, F
) is isometric and it is unitary if they are both isometrics.

We will denote by Ω a region in the polydisc and F Q stands for the direct sum F ⊕ Q.

By kernel we mean a function

K : Ω × Ω → L(F) (4.6)
for a Krein space F. It is Hermitian if K(z, w) = K(w, z), ∀z, w ∈ Ω.

One says that a Hermitian, L(F)-valued kernel K defined in Ω × Ω has k number of negative squares if for all z 1 , z 2 , . . . , z n ∈ Ω and f 1 , f 2 , . . . , f n ∈ F, n ∈ N, the Hermitian matrix

(⟨K(z i , z j )f i , f j ⟩) n i,j=1 (4.7)
has at most k negative eigenvalues and there exists at least one such matrix with exactly k negative eigenvalues.

Let now F be a functional Pontryagin space with coefficient space a Krein space K i.e a Pontryagin space of functions defined in Ω with values in K.

We have [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF]:

Definition 4.1. A kernel K : Ω × Ω → L(K) is said to be a reproducing kernel of F if ∀w ∈ Ω and f ∈ F:

1) K(w, z)f ∈ F as a function of z 2) ⟨h(•), K(w, •)f ⟩ F = ⟨h(w), f ⟩ F , ∀h ∈ F.
If such kernel exists, F is called a reproducing kernel Pontryagin space.

As it is well-known in reproducing kernels theory [START_REF] Aronszajn | Theory of reproducing kernels[END_REF][START_REF] Saitoh | Theory of reproducing kernels[END_REF][START_REF] Saitoh | Theory of reproducing kernels and applications[END_REF], such kernel exists if and only if all evaluation mappings act continuously from F into K.

Another useful result which will be used in the sequel is the following.

Theorem 4.1 (see [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF]). Let K be a Hermitian kernel defined on Ω × Ω with values in L(K), K a Krein space, and has a finite number of negative squares k. Then there exists a unique Pontryagin space F of K-valued functions on Ω with reproducing kernel K and negative index k.

Proof. See appendix B or [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF].

An operator colligation is a quadruple (H, F, Q, V ) consisting of three Krein spaces H, F, Q and a connecting operator

V ∈ L(H ⊕ F, H ⊕ Q). H
is the state space. So we have:

V = T F G H : H F -→ H Q (4.8)
where T ∈ L(H), F ∈ L(F, H), G ∈ L(H, Q) and H ∈ L(F, Q). We mean by characteristic function of the colligation (H, F, Q, V ) the function given by:

Θ(z) = H + G(I H -zT ) -1 zF (4.9)
The colligation (H, F, Q, V ) is said to be closely outer connected if

span{ranT * n G * , n ⩾ 0},
where ran denotes the range space, is dense in the state space H.

Realization of the Generalized Schur Functions

Let S : Ω -→ L(F, Q) be a holomorphic L(F, Q)-valued function, with F and Q Pontryagin spaces, defined on a region Ω of the polydisc D d containing the origin. One associates to S the kernel There is a unique partial coisometric colligation

K S (w, z) = I Q -S(z)S(w) 1 -⟨z, w⟩ (4
V = T F G H : H(S) F -→ H(S) d Q (4.12) with H(S) d = H(S) ⊗ • • • ⊗ H(S)
and such that, for all h ∈ H(S), f ∈ F and z = (z 1 , . . . , z d ) ∈ Ω(S), we have: Proof. See appendix B.

                     (zT h)(z) = h(z) -h(0) (zF f )(z) = (S(z) -S(0))f Gh = h(0) Hf = S(0)f (4.13) where T =     T 1 . . . T d     , F =     F 1 . . .

Realization of the Generalized Herglotz Functions

The other class in which we are interested is that of Herglotz. Let us first give the definition of this class.

K Φ (z, w) = Φ(z) + Φ(w) 1 -⟨z, w⟩ (4.15)
has a finite number of negative squares. For d = 1, the following theorem can be proved using the method of linear relations in exactly the same way as in [START_REF] Abu-Ghanem | Herglotz functions of several quaternionic variables[END_REF][START_REF] Alpay | Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions[END_REF].

Theorem 4.3. Let Φ ∈ H 0 (F).Then it can be represented as

Φ(z) = iℑ{Φ(0)} + 1 2 G(I L(Φ) -zT ) -1 (I L(Φ) + zT )G * (4.16)
where L(Φ) is the reproducing kernel Pontryagin space generated by the kernel K Φ , T is an isometry and G is the point evaluation map at the origin.

Proof. Consider the linear relation R ⊂ L(Φ) × L(Φ) spanned by the pairs given by:

(K Φ (z, α)αu, K Φ (z, α)u -K Φ (z, 0)u) (4.17)
with α ∈ Ω(Φ) and u ∈ F.

Following [START_REF] Abu-Ghanem | Herglotz functions of several quaternionic variables[END_REF][START_REF] Alpay | Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions[END_REF], the relation R is an isometry and therefore extends to the graph of a densely defined isometric operator

T : L(Φ) - → L(Φ).
Now we compute its adjoint T * . Let f ∈ L(Φ), we have

u * ᾱ(T * f )(α) = ⟨u * ᾱT * f, K Φ (z, α)⟩ (4.18) = ⟨f, K Φ (z, α)u -K Φ (z, 0)u⟩ = u * (f (α) -f (0)). (4.19)
which can be written as

(I L(Φ) -ᾱT * )f (α) = f (0). (4.20)
Applying this last relation to f

(α) = K Φ (z, α) K Φ (z, α) = (I L(Φ) -ᾱT * ) -1 K Φ (z, 0). (4.21)
Applying the mapping G and taking into account that K Φ (z, 0) = G * 1, we get:

K Φ (0, α) = Φ(α) * + Φ(0) (4.22) = G(I L(Φ) -ᾱT * ) -1 K Φ (z, 0) (4.23) = G(I L(Φ) -ᾱT * ) -1 G * . (4.24)
The rest is therefore a reorganization:

Φ(α) * = G(I L(Φ) -ᾱT * ) -1 G * - Φ(0) + Φ(0) * 2 - Φ(0) -Φ(0) * 2 = -iℑ(Φ(0)) + G(I L(Φ) -ᾱT * ) -1 G * - 1 2 GG * = -iℑ(Φ(0)) + 1 2 G(I L(Φ) -ᾱT * ) -1 (I L(Φ) + ᾱT * )G *
Applying the adjoint operation on both sides gives the desired equation (4.16).

The general case

We study the general case by using a standard technique in multivariable analysis, the so-called slice functions [START_REF] Rudin | Function theory in polydiscs[END_REF]. This allows to reduce, systemat-ically, the general case to the one variable in §4.4.1. The slices are already used in [START_REF] Serban | Schur coefficients in several variables[END_REF] to introduce the notion of functional Schur parameter. This technique will also allow here to leave the coefficient space F being a Krein space and so to proceed without any other condition like (4.

Φ ω (λ) = Φ(λω, λ) = Φ(λω 1 , λω 2 , . . . , λω d-1 , λ) (4.25)
Each slice Φ ω belongs, by definition, to the class H 0 (F). More compactly, we consider the application Γ mapping the unit circle

T d-1 of C d-1 into the class H 0 (F) Γ : T d-1 -→ H 0 (F) ω -→ Φ ω
We endow the class H 0 (F) by the following norm

∥Ψ∥ r = sup |λ|<r ∥Ψ(λ)∥ F , ∀Ψ ∈ H 0 (F), 0 < r < 1 (4.26)
where ∥ • ∥ F is the operator norm defined from the one which induces the strong topology in the Krein space F.

Proposition. The mapping Γ : T d-1 -→ H 0 (F) defined above is continuous.

Proof. Let ω, ω ′ ∈ T d-1 and Φ ∈ H d 0 (F). For each λ ∈ Ω(Φ), Φ ω (λ) is a continuous linear operator in the Krein space F, so there is a constant M ω,λ depending on ω and λ such that ∥Φ ω (λ)∥ F ⩽ M ω,λ . Let M = sup{M ω,λ , ω ∈ T d-1 , λ ∈ B}. By a recursive use of Cauchy formula, we have

∥Φ(z) -Φ(z ′ )∥ F ⩽ d∥Φ(•)∥ F ∥z -z ′ ∥ (1 -∥z∥)(1 -∥z ′ ∥)
where ∥z∥ = sup 1⩽i⩽d |z i |. Finally, using the definition of the slices, we get

∥Φ ω -Φ ω ′ ∥ r ⩽ (d -1)M ∥ω -ω ′ ∥ (1 -r) 2
showing that the mapping Γ is continuous.

Each slice Φ ω ∈ H 0 (F) of Φ ∈ H d 0 (F) can be represented as the realization equation (4.16). Moreover, there exists a reproducing kernel Pontryagin space L(Φ ω ) associated to each of them. These spaces are linked to the space L(Φ) by the following.

Proposition. Let Φ ∈ H d 0 (F).
The span of the union of the spaces L(Φ ω ), w ∈ T d-1 , is dense in the space L(Φ).

Proof. Take an f ∈ L(Φ) orthogonal to L(Φ ω 0 ), for some ω 0 ∈ T d-1 . So f is orthogonal to the kernel K Φ ω 0 (•, •) ∈ L(Φ ω 0 ). Use then the fact that

this kernel K Φ ω 0 (•, •) is easily expressible in terms of K Φ (•, •) ∈ L(Φ) as K Φ ω 0 (α, β) = K Φ (αω 0 , βω 0 ), α, β ∈ D.
Then, f ≡ 0 thanks to the reproducing property in L(Φ) and the proof ends.

The remaining part which is not solved in this dissertation is the way to get back to the function Φ(z) from its slices. One approach which we will explore in the future consists in comparing the generalized Herglotz function Φ(z) to the impedance of the system obtained by cascading the systems defined by the slices (4.25) through the realization equation (4.16).

Conclusion

We have proposed a clear cut definition of the scale-shift operator in discretetime. Its implementation has been given under a very simple classical filtering method. It has then been illustrated with some examples where, we have noted that finitely supported signals induce, under the scale-shift operator, Gibbs phenomenon. We have proposed a method to remove this phenomenon. Subsequently, this operator has been used to define rigorously and in an abstract harmonic analysis framework the discrete-time Fourier-Mellin transform. The atoms of the decomposition have been explicitly given and a fast computational algorithm has also been explained. The definition of this operator has enabled us to establish an explicit equivalence between two popular signal transforms: the voice transform of a specific Blaschke hyperbolic subgroup and the wavelet transform of a given discrete-time signal. We then further explored this equivalence and arrived at a sufficient condition to construct a complete wavelet family from a purely discrete-time signal. Another theme that has been addressed in this thesis is the realization problem for generalized Schur and Herglotz classes of functions. We have adapted, under a sufficient condition on indices of the coefficient spaces, the classical one-variable proof to the generalized Schur class. We then used the notion of slice function to reduce the problem, in the Herglotz case, from multivariables to a single variable. Some topological results have been provided. To finish the proof, we need to express the development of Φ n (s, z) in a formal power series of the variable z, s being a fixed parameter. To begin, we set

Conclusion

A s (z) := λ -s-1/2 z 1 + z = (1 -z) -s-1/2
(1 + z) -s+1/2 . The following formal power series n! (p k,2q (s) + p k,2q (-s)) + 1 (q!) 2 p q,2q (s).

-For n = 2q + 1, where the maximum is taken over all subspaces of the span of g 1 , . . . , g n .

P 2q+1 (s) = 1 n! 2q+1 k=0 2q + 1 k p k,2q+1 (s)
This gives the reverse equality and therefore equality holds. Then, by construction, P contains isometrically the spaces F 0 , F 0+ and F -and also F -is dense in P. Therefore, F 0+ is dense in P ⊖ F -and so P ⊖ F -coincides with F + constructed above. Thus P is isometrically equal to the space F. The formulas for T, F, G and H in (4.13) then follow.

The proof of (4.14) comes from the fact that the evaluation mapping at a given z ∈ Ω(S) is given by G(I -zT ) -1 .

Indeed, to see this, let h ∈ H(S) and let k ∈ H(S) be defined by k = (I -zT ) -1 h.

We just proved that

(zT k)(z) = k(z) -k(0)
which is equivalent to

[(I -zT )k](z) = k(0)
and saying that the value of h = (I -zT )k at the point z is given by k(0) = Gk (by (4.13)). Hence,

h(z) = Gk(z) = G(I -zT ) -1 h

  Mots-clés: Translation en échelle en temps discret, Groupe hyperbolique de Blaschke, Transformée de Mellin, Voice transform, Ondelettes, Réalisation de systèmes. CONTENTS 3 The Discrete Scale-Shift approach to Wavelet and Voice transforms 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Brief review of wavelet theory . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Voice transform . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 3 A. 1

 31 Voice transform to Wavelets coefficients . . . . . . . . . . . . . . . . . . . . 3.4 Discrete-time Wavelet family . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Realization of Generalized Schur and Herglotz Classes of Functions 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Back to the classic realization problem . . . . . . . . . . . . . . . . 4.2.2 Generalities on Kreins spaces . . . . . . . . . . . . . . . . . . . . . 4.3 Realization of the Generalized Schur Functions . . . . . . . . . . . . . . . . 4.4 Realization of the Generalized Herglotz Functions . . . . . . . . . . . . . . 4.4.1 The case d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion A Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Computation of M Φ (α) n (z) . . . . . . . . . . . . . . . . . . . . . A.1.2 Computation of ⟨z m , Φ n (s, z)⟩ . . . . . . . . . . . . . . . . . . . . . Chapter 0

  a causal discrete-time signal and let m ∈ N be a given time index. Then 1) -The Mellin transform of {f n } n⩾0 with respect to the time index m, denoted by f m = M m {f }, is given by:

  = d j=1 z j T j and zF = d j=1 z j F j . The colligation is closely outer connected and its characteristic function coincides with the function S:

  we use very classical digital filtering tools to calculate its scale shifted counterpart {f (α) n } n∈N , for an arbitrary scale α ∈ R + * . When the original signal has a finite support N , we show through a comprehensive example, that its scale-shifted version undergoes Gibbs oscillations whenever α > 1. Solution to avoid the Gibbs phenomenon is provided. In chapter 2, we use this discrete scale shift approach to give an exact and purely harmonic analysis definition of the Mellin transform for discrete-time signals. It is an atomic representation and a computational algorithm is derived which shows how easy and fast these atoms can be computed recursively. In chapter 3, we use the mathematics of the scale-shift operator and consider unitary representations of the hyperbolic Blaschke group in the Hardy space H 2 (D) of the unit disc. We then establish that the discrete wavelet transform is mathematically equivalent to the Voice transform for an Abelian subgroup of the hyperbolic Blaschke group. This strict link is then exploited to furnish an admissible condition which is sufficient to construct complete wavelet families from discrete-time signals. In chapter 4, which can be read separately from the three chapters above, we use the well-know notion of slice function Introduction Dans ce chapitre, nous étudions profondément la définition de l'opérateur de translation en échelle pour les signaux à temps discret. Tous les détails mathématiques utilisés pour aboutir à une définition claire de cet opérateur sont exposés. Dans la section 2, nous discutons ces détails techniques à travers une analogie avec le cas du temps continue qui est plus facile à élaborer. Cette analogie nous permettra d'interpréter cet opérateur comme une action d'un groupe. Dans la section 3, son implémentation est étudiée et donnée dans un cadre très simple et très classique de filtrage digitale. Dans la section 4, nous allons voir, à travers un exemple simple, comment les oscillations de Gibbs viennent corrompre les versions translatées en échelle des signaux à support fini. Une solution pour supprimer ces oscillations est ensuite proposée.

n

  (z)} n⩾0 is orthogonal and complete in the Hardy space H 2 (D) of the unit disc. Hence, expressing each Φ (γ) n (z) in the standard basis of H 2 (D), we can rewrite T α γ F (z) as

Figure 1 . 1 :

 11 Figure 1.1: Implantation diagram obtained from (1.15), (1.22) and (1.23)

  figure provides an illustration of the composition property(1.11):α γ -1 = 1 α γ

√ 2

 2 for the red curve and α = 1 β = √ 2 for the blue curve.

Figure 1 . 2 :

 12 Figure 1.2: Effectiveness of the scale shift operator even for irrational scales (top plot) -Reversibilty of the scaling (bottom plot)

1 √ 3 .

 13 For example, N = 345 satisfies this condition, with M = 255. The jump in the original signal, displayed by the black colored curve in figure 1.3, induces Gibbs oscillations for all scales α 1 = √ 3 and α 2 = With α 1 > 1, the oscillations occur in the interval [n α 1 , N α 1 ], as claimed above (see the blue colored curve in figure 1.3).

Figure 1 . 3 :

 13 Figure 1.3: Gibbs oscillations and scale shift of a nonsmooth signal.

Figure 1 . 4 : 3 .

 143 Figure 1.4: Making scale shitf insensitive to Gibbs oscillations

Figure 1 . 5 :

 15 Figure 1.5: Scale-shift immune to Gibbs oscillations

  Dans ce chapitre, en utilisant l'opérateur discret de translation en échelle, nous donnons une définition de la transformée de Mellin en temps discret.Cette définition est exacte et pûrement de l'analyse harmonique. La transformée de Mellin est un outil très connu en Mathématiques pour son utilisation dans la résolution d'équations fonctionnelles, dans l'analyse d'algorithmes, etc. Son utilité découle de ses importantes propriétés basiques qui sont très faciles à établir mais aussi de ses propriétés asymptotiques remarquables.

2. 1 . 2

 12 Basic definitions Definition 2.1. A function σ : G → T, T being the unit circle, satisfying

m

  are therefore two unrelated functions of α. Hence a different Mellin transform needs to be computed for each different m. We have [59] Theorem 2.1. Let f = {f n } n⩾0 ∈ ℓ 2 be a causal discrete-time signal and let m ∈ N be a given time index. Then 1) -The Mellin transform of {f n } n⩾0 with respect to the time index m, denoted by f m = M m {f }, is given by

Figure 2 . 1 :

 21 Figure 2.1: Some atoms E m,n of the Mellin transform

Theorem 3 . 1 .

 31 The reason lies probably on the lack of a proper scale-shift operator. Indeed, because of the non commutativity of the whole Blaschke group G, different elements of the group present different rotation angles and this precludes to interpret the action of the entire group in terms of scale-shift. Here, we get around this difficulty by fixing the angle β and just consider the subgroup G β of SU (1)which, in turn, is Abelian. Definition 3.1 and expression (3.6) lead to the following result[START_REF] Ngom | Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals[END_REF]: The voice transform of a discrete-time signal {f n } n⩾0 is equivalent to its discrete wavelet transform as given in(3.6), that is:

Figure 3 . 1 :Figure 3 . 2 :

 3132 Figure 3.1: A signal with a rupture at approximately 513 ms

  1) ℓ-m g k+m This last equation interprets the wavelet coefficients of the signal g = {g n } n as a difference equation of itself around time k at fine scales analogously to (3.40) in continuous time. With the same previous signal example, we calculate its wavelet coefficients by taking as wavelet the one given in (3.42) where the process {θ n } n is a Gaussian white noise and ℓ = 1. High amplitude coefficients follow parabolas and move closer to the break location at fine scales (see figure below). The process f = {f n } n , obtained in this way, can be interpret as a wavelet with ℓ vanishing moments and it is also, as the process θ = {θ n } n , a white Gaussian noise.

Figure 3 . 3 :

 33 Figure 3.3: Wavelet coefficients computed with white Gaussian noise.

  Dans ce chapitre, nous nous intéressons au problème de réalisation pour les classes de fonctions généralisées de Schur et de Herglotz. Ces classes sont constituées de fonctions définies dans le polydisc, à valeurs opérateurs définis dans des espaces de Pontryagin et dont les noyaux associés admettent une certaine négativité. Dans la section 2, nous revisitons le problème de réalisation sous sa forme classique puis passons en revue des espaces de Pontryagin.Ces espaces sont des cas particuliers d'espaces plus généraux, les espaces de Krein. En section 3, le cas de l'espace de Schur généralisé est considéré. Sous une condition suffisante sur les indices des espaces coefficients, nous donnons la réalisation en adaptant les techniques classiques utilisées pour ce même problème mais dans le cas d'une variable. Dans la section 4, nous étudions le cas de l'espace généralisé de Herglotz. Nous donnons d'abord la réalisation dans le cas d'une variable dont la preuve peut être déduite dans de nombreux travaux tels que[START_REF] Abu-Ghanem | Herglotz functions of several quaternionic variables[END_REF][START_REF] Alpay | Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions[END_REF][START_REF] Ball | Schur-Agler and Herglotz-Agler classes of functions: positive-kernel decompositions and transfer-function realizations[END_REF]. Pour le cas général, nous contournons le problème en attachant des fonctions de tranche à chaque fonction de Herglotz pour se ramener, systématiquement, au cas plus simple d'une seule variable. Nous présentons quelques résultats topologiques. Noter que la notion de tranche est standard en analyse multivariable[START_REF] Rudin | Function theory in polydiscs[END_REF].

4. 1 Introduction

 1 In this chapter, we are interested to the realization problem for the generalized Schur and Herglotz classes of functions. These classes consist of functions defined in the polydisc which are operator-valued in Pontryagin spaces and whose the associated kernels admit finite numbers of negative squares. In section 2, we revisit the realization problem in its classical form and then review briefly the theory of Pontryagin spaces. These spaces are special particular class of the more general class of Krein spaces. In section 3, the case of the generalized Schur class is considered. Under a sufficient condition, the problem can be solved by adapting the classical techniques used for the single

  introduced the class of analytic functions S defined in a domain of the polydisc D d and which satisfy: |S(rT 1 , . . . , rT d )| ≤ 1 (4.1) for any r < 1 and for any d-tuple of commuting contractions (T 1 , . . . , T d ) on a Hilbert space H(S). This class named Schur-Agler is then characterized in the sense that (4.1) is equivalent to the existence of holomorphic operatorvalued functions H k , k = 1, . . . , d, defined in D d such that, for any z = (z 1 , . . . , z d ) and w = (w 1 , . . . , w d ) in the domain of definition of S, we have:

  decomposition H(S) = H 1 ⊕ • • • ⊕ H d of the Hilbert space H(S) such that:

  is the Euclidean inner product and I Q is the identity element of the space Q.The generalized Schur class in which we are interested here is the set of all those functions S for which the kernel (4.10) has a finite number of negative squares denoted by k. Since, by definition, the kernel (4.10) is also hermitian, then by Theorem 4.1, there exists a unique reproducing kernel Pontryagin space H(S) with K S as reproducing kernel and which has negative index k. If the negative indices of the coefficient spaces F and Q satisfy the condition ind -F = ind -Q + (d -1)k (4.11) then the following result holds Theorem 4.2.

  = d j=1 z j T j and zF = d j=1 z j F j . The colligation is closely outer connected and its characteristic function coincides with the function S S(z) = H + G(I H(S) -zT ) -1 zF (4.14)

Definition 4 . 2 .

 42 The generalized Herglotz class of functions denoted by H d 0 (F), d ⩾ 1, F a Krein space, is the set of all holomorphic L(F)-valued functions Φ defined in a domain Ω(Φ) ⊂ D d containing the origin and such that the kernel:

4. 4 . 1

 41 The case d = 1

  11) which is necessary to directly study the problem as in the Schur case §4.3. Our plan is first to study if it possible to continuously attach its slices to a given generalized Herglotz function. If so, the result of Theorem 4.3 can be applied to each slice and, finally, the remaining task is how to get back to the source Herglotz function. Now, let Φ ∈ H d 0 (F) and let T d-1 denotes the unit circle in C d-1 . To each ω = (ω 1 , ω 2 , . . . , ω d-1 ) ∈ T d-1 , one associates the following slice function Φ ω defined in the disc D and given by:

A. 1 . 2

 12 Nous avons proposé une définition claire et théorique de l'opérateur de translation en échelle en temps discret. Son implémentation a été donnée sous une méthode classique très simple de filtrage numérique. Nous l'avons ensuite illustré avec quelques exemples où nous avons constaté que les signaux à support fini induisent, sous l'opérateur de translation en échelle, le phénomène de Gibbs. Nous avons proposé une méthode pour supprimer ce phénomène. Dans la suite, cet opérateur a été utilisé pour définir rigoureusement et dans un cadre abstrait de l'analyse harmonique la transformée de Mellin en temps discret. Les atomes de la décomposition ont été données explicitement et un algorithme de calcul rapide a aussi été expliqué. La définition de cet opérateur nous a également permis d'établir une équivalence stricte entre deux transformées populaires en théorie du signal: le voice transform d'un sous groupe spécifique du groupe de Blaschke et la transformée en ondelettes d'un signal en temps discret. Cette équivalence a ensuite été explorée davantage pour arriver à une condition suffisante pour contruire une famille d'ondelettes complète à partir d'un signal purement à temps discret. Un autre thème abordé dans cette thèse est le problème de réalisation pour les classes de fonctions généralisées de Schur et de Herglotz dans les espaces de Pontryagin. Sous une condition sur les indices des espaces coefficients, nous avons pu adapter les techniques classiques de la preuve de ce problème en une variable dans ce cadre général pour la classe de Schur. Pour la classe généralisée de Herglotz, nous avons fait appel aux fonctions de tranches pour réduire le cas général au cas plus simple d'une seule variable. Quelques ré-sultats topologiques ont été fournis. and recalling the expression (2.13), namely p k,n (s) = k,n (s) n! sin [π(s -k -1Computation of ⟨z m , Φ n (s, z)⟩

( 1 -

 1 z) -s-1/2 = ∞ ℓ=0 Γ(s + 1/2) Γ(s -ℓ + 1/2)Γ(ℓ + 1) z ℓ (1 + z) -s+1/2 = ∞ ℓ=0 (-1) ℓ Γ(-s + 1/2) Γ(-s -ℓ + 1/2)Γ(ℓ + 1)z ℓ , P n (s), for n even/odd, by:-For n = 2q,

  ,n (s) + p n-k,n (s)) ,n (s) -p k,n (-s)) .The Lemma follows upon noting that for s = iω, ω ∈ R, it holds thatp k,n (-iω) = p k,n (iω) since all coefficients of p k,n are real. □ have, for , max{⟨Gy, y⟩ C n , y ∈ span{x 1 , . . . , x r }, ∥y∥ = 1} < 0 (B.7)We also have by invoking the minimax principleλ r = min max{⟨Gy, y⟩ C n , y ∈ L r ∥y∥ = 1} (B.8)where the minimum is taken over all r-dimensional subspaces L r of C n . Thus λ r ⩽ 0 and so follows ν(G) ⩾ max dim n (B.9)

  Now we come back to the proof of the existence in Theorem 4.1. LetF 0 = Span{K(w, •)f, w ∈ Ω, f ∈ K} (B.10)Endow F 0 with an inner product ⟨•, •⟩ F 0 by requiring thatn j=1 K(w j , •)f j , j , w i )f j , f i ⟩ K (B.11)for w 1 , . . . , w n ∈ Ω and f 1 , . . . , f n ∈ K. ⟨•, •⟩ F 0 is well defined and for all h ∈ F 0 and f ∈ K, we have⟨h(•), K(w, •)f ⟩ F 0 = ⟨h(w), f ⟩ K (B.12)If now the kernel K has a finite number of negative squares k then there exist α 1 , . . . , α n ∈ Ω and f 1 , . . . , f n ∈ K such that the Gram matrix(⟨K(α j , •)f j , K(α i , •)f j ⟩) n i,j=1 = (⟨K(α j , α i )f j , f i ⟩) n i,j=1(B.13) has k negative eigenvalues. Then by Lemma 4, F 0 contains a maximal kdimensional subspace F -which is the antispace of a Hilbert space. Now we B.1.2 The uniqueness part Let P be any Pontryagin space of function on Ω with reproducing kernel K(w, z).

B. 2 F 1 u 2 ,where u 1 2 , 2 , 2 , 2 H+ 1 u 2 ,FFQF 1 u 2 H

 212122221212 Proof of Theorem 4.2 Consider the linear relation R ⊆ H d (S) spanned by the pairsα * K S (α, z)u [K S (α, z) -K S (0, z)]u 1 + K S (0, z)u 2 [S(α) * -S(0) * ]u 1 + S(0) * u 2where H d (S) ⊆ H(S) d is the closed spanned of the elementsα * K S (α, z)u 1 = K S (α, z)u 1 ᾱ2 K S (α, z)u 1 . . . ᾱd K S (α, z)u 1 ∈ Q, and α = (α 1 , . . . , α d ) ∈ Ω(S).The relation R is therefore densily defined inH d (S)Q .In order to apply Schmulyan theorem on the extension of linear relations in Pontryagin spaces, we should further show that R is isometric.Letβ * K S (β, z)v 1 v [K S (β, z) -K S (0, z)]v 1 + K S (0, z)v 2 [S(β) * -S(0) * ]v 1 + S(0) * v 2 where v 1 , v 2 ∈ Q, β = (β 1 , . . . , β d ) ∈ Ω(S), be another pair in R.We have[K S (α, z) -K S (0, z)]u 1 + K S (0, z)u 2 [S(α) * -S(0) * ]u 1 + S(0) * u [K S (β, z) -K S (0, z)]v 1 + K S (0, z)v 2 [S(β) * -S(0) * ]v 1 + S(0) * v 2 H(S)⊕F = ⟨K S (α, β)u 1 , v 1 ⟩ Q -⟨K S (α, 0)u 1 , v 1 )⟩ Q + ⟨K S (α, 0)u 1 , v 2 ⟩ Q --⟨K S (0, β)u 1 , v 1 ⟩ Q + ⟨K S (0, 0)u 1 , v 1 ⟩ Q -⟨K S (0, 0)u 1 , v 2 ⟩ Q + + ⟨K S (0, β)u 2 , v 1 ⟩ Q -⟨K S (0, 0)u 2 , v 1 ⟩ Q + ⟨K S (0, 0)u 2 , v 2 ⟩ Q + ⟨[S(β) -S(0)][S(α) * -S(0) * ]u 1 , v 1 ⟩ Q + ⟨S(0)S(0) * u 2 , v 2 ⟩ Q + + ⟨[S(β) -S(0)]S(0) * u 2 , v 1 ⟩ Q + ⟨S(0)[S(α) * -S(0) * ]u 1 , v 2 ⟩ Q = d i=0 ᾱi β i ⟨K S (α, β)u 1 , v 1 ⟩ Q + ⟨u 2 , v 2 ⟩ Q = d i=0 ⟨ᾱ i K S (α, z)u 1 , βi K S (β, z)v 1 ⟩ Q + ⟨u 2 , v 2 ⟩ QWe continue the computation[K S (α, z) -K S (0, z)]u 1 + K S (0, z)u 2 [S(α) * -S(0) * ]u 1 + S(0) * u [K S (β, z) -K S (0, z)]v 1 + K S (0, z)v 2 [S(β) * -S(0) * ]v 1 + S(0) * v K S (α, z)u 1 ᾱ2 K S (α, z)u 1 . . . ᾱd K S (α, z)u 1 K S (β, z)v 1 β2 K S (β, z)v 1 . . . βd K S (β, z)v 1 ⟨u 2 , v 2 ⟩ Q = α * K S (α, z)u that R is an isometric linear relation in H d (S) negative index.Therefore, there is an isometryV * = T * G *whose graph is equal to the linear relation R. To find the formulas for T, F, G and H in (4.13) let h ∈ H(S), f ∈ F, g ∈ Q such that Hence, for u 1 , u 2 ∈ Q, and α = (α 1 , . . . , α d ) ∈ Ω(S), we have ⟨αk(α), u 1 ⟩ Q + ⟨g, u 2 ⟩ Q = k* H * α * K S (α, •)u (α, •) -K S (0, •)]u 1 + K S (0, •)u 2 [S(α) * -S(0) * ]u 1 + S(0) * u 2 = ⟨[h(α) -h(0)], u 1 ⟩ Q + ⟨h(0), u 2 ⟩ Q + + ⟨[S(α) -S(0)]f, u 1 ⟩ Q + ⟨S(0)f, u 2 ⟩ Q

  based on the knowledge of h τ (σ) for any given pair (τ, t) ∈ E 2 . This arises when the group action is transitive. Recall that the action of G on E is said transitive if for any pair (τ, t) ∈ E 2 , there exists a ∈ G such that τ = η(a, t). Let then t 0 ∈ E be fixed once for all and given t ∈ E, let b t ∈ G be defined such that η(b t , t 0 ) = t. Then we get:

	t)	(2.5)
	Now, for each fixed t ∈ E, one can compute by (2.3) the Fourier transform
	h

t (σ) of the function h t (•) = h(•, t), provided it belongs to L 1 (G , dµ). In general, one has to compute h t (σ) via (2.3) for each different t. In some situations however, it becomes straightforward to determine h t (σ)

1.4. FINITE SUPPORT AND GIBBS PHENOMENON

Acknowledgments

This work is dedicated to the memory of my Dad, Mamadou NGOM, who left us a few months before the end of this journey.

Chapter 2

The Discrete Scale-Shift approach to the Mellin Transform

n (z) (s)

The first task towards the computation of M Φ (α)

n (z) as a function of α, regarding n and z as fixed parameters. Without loss of generality, we fix θ = 0 and consider the subgroup G 0 . The dependence of

n (z) on α can be made explicit by recalling (1.16), (1.12) and (1.13) with θ = 0. We get

where we have set

and where we have used the well-known binomial formula in the last equality.

Next, recall that if Y (s) represents the Mellin transform of a function y(x)

then the Mellin transform of x ν y(x) would read as Y (s + ν). So, let us set

The corresponding Mellin transform is well-known as

where Γ(•) is the classical Gamma function. Therefore, we deduce the Mellin transform

.

With the Euler's reflection formula

where

with r 0 (s) ≡ 1 and

Hence, replacing k by m -k and s by -s, leads to

If we split the product in the expression of p k,n (s) in (2.13) as

then one immediately recognizes the equality

This leads to the conclusion that

The proof is now complete with

Evaluating the atoms E m,n (s) on the imaginary axis i.e s = iω, ω ∈ R, gives

which is exactly (2.10).

A.

= (-1) n p k,n (-s)

where

Observe first that the symmetry relation (2.14) implies that p q,2q (iω) = p q,2q (-iω) = p q,2q (iω) The proof of the existence use the following (see [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF]).

Consider their Gram matrix G = (⟨g i , g j ⟩ h ) n i,j=1 . Then the number of negative eigenvalues of G coincides with the maximum dimension of a subspace n of the span of g 1 , • • • , g n which is an antispace of a Hilbert space in the inner product ⟨•, •⟩ h of h.

Proof. Let λ 1 ⩽ • • • ⩽ λ n the eigenvalues of the matrix G and that there is ν(G) number of them which are negative. Let 1 ⩽ r ⩽ ν(G), then λ r ⩽ 0 and G has orthonormal eigenvectors

corresponding to the eigenvalues λ 1 , . . . , λ r . Consider the family of vectors (u j , j = 1, . . . , r) given by

For all i, j = 1, . . . , r, we have

Therefore, u 1 , . . . , u r span an r-dimensional subspace n of the span of g 1 , . . . , g n which is the antispace of a Hilbert space in the inner product of h and so

where max is taken over all such spaces. It remains to prove the reverse inequality.

Let n be an r-dimensional subspace of the span of g 1 , . . . , g n which is the antispace of a Hilbert space in the inner product of h. Choose v 1 , . . . , v r ∈ n such that

Then there exist numbers γ ip , i = 1, . . . , r, p = 1, . . . , n such that

Now define x 1 , . . . , x r from the γ ip 's as in (B.1). We can see that

Consider the kernel defined by

for all w, z ∈ Ω and f ∈ K.

Define another subspace F 0+ equals to the span of the functions K + (w, •)f with w ∈ Ω, f ∈ K and where

The subspaces F 0+ and F -are orthogonal since

for all l = 1, . . . , k.

It remains now to prove that F 0+ can be completed to a reproducing kernel hiblert space with reproducing kernel K + (w, z). For that, it suffices to prove

for w 1 , . . . , w n ∈ Ω and f 1 , . . . , f n , g 1 , . . . , g n ∈ K.

We have

⟨K + (w j , w i )f j , g i ⟩ K

Now, by contradiction, suppose that K + (w, z) is not nonnegative. Then there exist β 1 , . . . , β q ∈ Ω and ψ 1 , . . . , ψ q ∈ K such that the element

Then the span of F -union with the element u k+1 is a (k + 1)-dimensional subspace of F 0 antispace of a Hilbert space which contradicts the maximality of F -. So K + (w, z) is nonnegative and thus F 0+ can be completed to a reproducing kernel Hilbert space F + with kernel K + (w, z).

Finally, define a Pontryagin space F = F + ⊕ F -. Since F ± have reproducing kernel K ± (w, z), F has reproducing kernel

In particular, take h = zF f with f ∈ F, z ∈ Ω(S). We have

and (4.14) follows.

For the uniqueness part, consider any colligation (4.12) satisfying (4.13).

From the first equation in (4.13), we have for h ∈ H(S), w ∈ Ω(S)

Likewise, we have

Therefore this colligation coincides with the one constructed above using linear relation.

List of Figures