Nizar Amri

Léonard Demni

Chaabane Gallardo

Margit Rejeb

Simon Rösler

Patrice Ruijsenaars

Michael Sawyer

AN INTRODUCTION TO DUNKL THEORY AND ITS ANALYTIC ASPECTS

Keywords:

q ż `8 0 dr r n´1 f prq jn´2 2 piλrq

(3) and f prq "

) involve now the modified Bessel function jn´2

2 , which can be characterized again in various ways :

' Relation with classical special functions and power series expansion . For every z P C,

q " e ´z 1 F 1 p n´1 2 ; n´1; 2zq , where J ν denotes the classical Bessel function of the first kind and p F q pa 1 , . . . , a p ; b 1 , . . .

Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was closely connected with certain classes of special functions in one variable :

' Bessel functions in connection with radial Fourier analysis on Euclidean spaces, ' Jacobi polynomials in connection with radial Fourier analysis on spheres, ' Jacobi functions (i.e. the Gauss hypergeometric function 2 F 1 ) in connection with radial Fourier analysis on hyperbolic spaces.

See [START_REF] Koornwinder | Jacobi functions and analysis on noncompact semisimple Lie groups[END_REF] for a survey. During the eighties, several attempts were made, mainly by the Dutch school (Koornwinder, Heckman, Opdam), to extend these results in higher rank (i.e. in several variables), until the discovery of Dunkl operators in the rational case and Cherednik operators in the trigonometric case. Together with q-special functions introduced by Macdonald, this has led to a beautiful theory, developed by several authors which encompasses in a unified way harmonic analysis on all Riemannian symmetric spaces and spherical functions thereon :

' generalized Bessel functions on flat symmetric spaces, and their asymmetric version, known as the Dunkl kernel, ' Heckman-Opdam hypergeometric functions on positively or negatively curved symmetric spaces, and their asymmetric version, due to Opdam, ' Macdonald polynomials on affine buildings.

Beside Fourier analysis and special functions, this theory has also deep and fruitful interactions with ' algebra (double affine Hecke algebras), ' mathematical physics (Calogero-Moser-Sutherland models, quantum many body problems), ' probability theory (Feller processes with jumps).

There are already several surveys about Dunkl theory available in the literature : ' [START_REF] Rösler | Dunkl operators (theory and applications)[END_REF] (see also [START_REF] Dunkl | Orthogonal polynomials of several variables[END_REF]) about rational Dunkl theory (state of the art in 2002), ' [START_REF] Opdam | Lecture notes on Dunkl operators for real and complex reflection groups[END_REF] about trigonometric Dunkl theory (state of the art in 1998), ' [START_REF] Etingof | Calogero-Moser systems and representation theory[END_REF] about integrable systems related to Dunkl theory, ' [START_REF] Macdonald | Affine Hecke algebras and orthogonal polynomials[END_REF] and [START_REF] Cherednik | Double affine Hecke algebras[END_REF] about q -Dunkl theory and affine Hecke algebras, ' [START_REF]Harmonic and stochastic analysis of Dunkl processes[END_REF] about probabilistic aspects of Dunkl theory (state of the art in 2006).

These lectures are intended to give an overview of some analytic aspects of Dunkl theory. The topics are indicated in red in Figure 1, where we have tried to summarize relations between several theories of special functions, which were alluded to above, and where arrows mean limits.

Let us describe the content of our notes. In Section 2, we consider several geometric settings (Euclidean spaces, spheres, hyperbolic spaces, homogeneous trees, . . . ) where radial Fourier analysis is available and can be applied successfully, for instance to study evolutions equations (heat equation, wave equation, Schrödinger equation, . . . ). Section 3 is devoted to the rational Dunkl theory and Section 4 to the trigonometric Dunkl theory. In both cases, we first review the basics and next address some important analytic issues. We conclude with an appendix about root systems and with a comprehensive bibliography. For lack of time and competence, we haven't touched upon other aforementioned aspects of Dunkl theory, for which we refer to the bibliography. The cosine functions ϕ λ pxq " cos λx (λ P C) occurring in these expressions can be characterized in various ways. Let us mention ' Power series expansion :

ϕ λ pxq " ÿ `8 ℓ"0 p´1q ℓ p2ℓq! pλxq 2ℓ @ λ, x P C .
' Differential equation : the functions ϕ " ϕ λ are the smooth eigenfunctions of p B Bx q 2 , which are even and normalized by ϕp0q " 1. ' Functional equation : the functions ϕ " ϕ λ are the nonzero continuous functions on R which satisfy ϕpx`yq `ϕpx´yq 2

" ϕpxq ϕpyq @ x, y P R.

2.2.

Hankel transform on Euclidean spaces. The Fourier transform on R n and its inverse are given by p f pξq "

ż R n dx f pxq e ´ixξ,xy (1)
and

f pxq " p2πq ´nż R n dλ p f pξq e ixξ,xy (2)
Notice that the Fourier transform of a radial function f " f prq on R n is again a radial function p f " p f pλq. In this case, (1) and (2) become

p f pλq " 2 π n 2
the generalized hypergeometric function. ' Differential equations . The function ϕ λ prq " jn´2 2 piλrq is the unique smooth solution to the differential equation

`B B r ˘2ϕ λ `n´1 r `B B r ˘ϕλ `λ2 ϕ λ " 0 ,
which is normalized by ϕ λ p0q " 1. Equivalently, the function

x Þ ÝÑ ϕ λ p|x|q " jn´2 2 piλ|x|q (5)
is the unique smooth radial normalized eigenfunction of the Euclidean Laplacian

∆ R n " ÿ n j"1 `B B x j ˘2 " `B B r ˘2`n ´1 r `B B r ˘`1 r 2 ∆ S n´1
corresponding to the eigenvalue ´λ2 .

Remark 2.1. The function (5) is a matrix coefficient of a continuous unitary representation of the Euclidean motion group R n ¸Opnq.

The function ( 5) is a spherical average of plane waves. Specifically,

ϕ λ p|x|q " ż Opnq dk e iλxu,k.xy " Γp n 2 q 2 π n 2 ż S n´1 dv e iλxv,xy ,
where u is any unit vector in R n . Hence the integral representation

ϕ λ prq " Γp n 2 q ? π Γp n´1 2 q ż π 0 dθ psin θq n´2 e iλ r cos θ " 2 Γp n 2 q ? π Γp n´1 2 q r 2´n ż r 0 ds pr 2 ´s2 q n´3 2 cos λs . ( 6
)
0 e 0 x 0 R n K-orbit Figure 2.
Real sphere S n 2.3. Spherical Fourier analysis on real spheres. Real spheres

S n " t x " px 0 , x 1 , . . . , x n q P R 1`n | |x| 2 " x 2 0 `. . . `x2
n " 1 u of dimension n 2 are the simplest examples of Riemannian symmetric spaces of compact type. They are simply connected Riemannian manifolds, with constant positive sectional curvature. The Riemannian structure on S n is induced by the Euclidean metric in R 1`n , restricted to the tangent bundle of S n , and the Laplacian on S n is given by ∆f " r ∆ r f ˇˇS n , where r ∆ " ÿ n j"0 `B B x j ˘2 denotes the Euclidean Laplacian in R 1`n and r f pxq " f `x |x| ˘the homogeneous extension of f to R 1`n t0u. In spherical coordinates $ ' ' ' ' ' ' & ' ' ' ' ' ' %

x 0 " cos θ 1 , x 1 " sin θ 1 cos θ 2 , . . .

x n´1 " sin θ 1 sin θ 2 . . . sin θ n´1 cos θ n , x n " sin θ 1 sin θ 2 . . . sin θ n´1 sin θ n ,
the Riemannian metric, the Riemannian volume and the Laplacian read respectively

ds 2 " ÿ n j"1 psin θ 1 q 2 . . . psin θ j´1 q 2 pdθ j q 2 , d vol " psin θ 1 q n´1 . . . psin θ n´1 qdθ 1 . . . dθ n and ∆ " ÿ n j"1 1 psin θ 1 q 2 ... psin θ j´1 q 2 `B B θ j ˘2`pn ´jqpcot θ j q B B θ j ( .
Let G " Opn `1q be the isometry group of S n and let K « Opnq be the stabilizer of e 0 " p1, 0, . . . , 0q. Then S n can be realized as the homogeneous space G{K. As usual, we identify right-K-invariant functions on G with functions on S n , and bi-K-invariant functions on G with radial functions on S n i.e. functions on S n which depend only on x 0 " cos θ 1 . For such functions, ż

S n d vol f " 2 π n 2 Γp n 2 q ż π 0 dθ 1 psin θ 1 q n´1 f pcos θ 1 q and ∆f " B 2 f B θ 2 1 `pn ´1qpcot θ 1 q Bf B θ 1 .
The spherical functions on S n are the smooth normalized radial eigenfunctions of the Laplacian on S n . Specifically, # ∆ ϕ ℓ " ´ℓ pℓ `n ´1q ϕ ℓ , ϕ ℓ pe 0 q " 1 , where ℓ P N. They can be expressed in terms of classical special functions, namely

ϕ ℓ px 0 q " ℓ ! pn´2q! pℓ`n´2q! C p n´1 2 q ℓ px 0 q " ℓ ! p n 2 q ℓ P p n 2 ´1, n 2 ´1q ℓ px 0 q or ϕ ℓ pcos θ 1 q " 2 F 1 `´ℓ, ℓ `n ´1; n 2 ; sin 2 θ 1 ˘,
where C pλq ℓ are the Gegenbauer or ultraspherical polynomials, P pα,βq ℓ the Jacobi polynomials and 2 F 1 the Gauss hypergeometric function.

Remark 2.2. We have emphasized the characterization of spherical functions on S n by a differential equation. Here are other characterizations :

' The spherical functions are the continuous bi-K-invariant functions ϕ on G which satisfy the functional equation ż K dk ϕpxkyq " ϕpxq ϕpyq @ x, y P G.

' The spherical functions are the continuous bi-K-invariant functions ϕ on G such that

f Þ ÝÑ ż G dx f pxq ϕpxq (8)
defines a character of the (commutative) convolution algebra C c pKzG{Kq. ' The spherical functions are the matrix coefficients ϕpxq " xπpxqv, vy , where π is a continuous unitary representation of G, which has nonzero K-fixed vectors and which is irreducible, and v is a K-fixed vector, which is normalized by |v| " 1. ' Integral representation :

ϕ ℓ pcos θ 1 q " Γp n 2 q ? π Γp n´1 2 q ż π 0 dθ 2 psin θ 2 q n´2 " cos θ 1 `i psin θ 1 qpcos θ 2 q ‰ ℓ " Γp n 2 q ? π Γp n´1 2 q psin θ 1 q 2´n ż sin θ 1 ´sin θ 1 ds psin 2 θ 1 ´s2 q n´3 2 pcos θ 1 `i sq ℓ . ( 9
)
The spherical Fourier expansion of radial functions on S n reads x j "

f pxq " ÿ ℓPN d ℓ xf, ϕ ℓ y ϕ ℓ pxq , where d ℓ " n pn`2ℓ´1q pn`ℓ´2q! n ! ℓ ! and xf, ϕ ℓ y " Γp n`1 2 q 2 π n`1 2 ż S n dx f pxq ϕ ℓ pxq " Γp n`1 2 q ? π Γp n 2 q ż π 0 dθ 1 psin θ 1 q n´1 f pcos θ 1 q ϕ ℓ pcos θ 1 q . 0 e 0 e n e j K-orbit N-orbit A-orbit Figure 3.
y j yn " 2z j 1 ´|z| 2 pj " 1, . . . , n´1q x n " 1 ´|y| 2 2yn " 2zn 1 ´|z| 2 # y j " x j x 0 `xn " 2z j 1 `|z| 2 `2zn pj " 1, . . . , n´1q y n " 1 x 0 `xn " 1 ´|z| 2 1 `|z| 2 `2zn # z j " x j 1 `x0 " 2y j 1 `|y| 2 `2yn pj " 1, . . . , n´1q z n " xn 1 `x0 " 1 ´|y| 2 1 `|y| 2 `2yn .
Let G be the isometry group of H n and let K be the stabilizer of a base point in H n . Then H n can be realized as the homogeneous space G{K. In Model 1, G is made up of two among the four connected components of the Lorentz group Op1, nq, and the stabilizer of e 0 is K " Opnq. Consider the subgroup A « R in G consisting of ' the matrices ' the Iwasawa decomposition G " NAK, which corresponds to Cartesian coordinates in Model 2. We shall denote by a rpgq and a hpgq the A `and A components of g P G in the Cartan and Iwasawa decompositions.

Remark 2.4. In small dimensions, H 2 « SLp2, Rq{ SOp2q and H 3 « SLp2, Cq{ SUp2q.

As usual, we identify right-K-invariant functions on G with functions on H n , and bi -K-invariant functions on G with radial functions on H n i.e. functions on H n which depend only on the distance r to the origin. For radial functions f " f prq, ż

H n d vol f " 2 π n 2
Γp n 2 q ż `8 0 dr psinh rq n´1 f prq and ∆f " B 2 f B r 2 `pn ´1qpcoth rq Bf B r . The spherical functions ϕ λ are the smooth normalized radial eigenfunctions of the Laplacian on H n . Specifically,

# ∆ ϕ λ " ´ λ 2 `ρ2 ( ϕ λ , ϕ λ p0q " 1 ,
where ρ " n´1 2 . Remark 2.5. The spherical functions on H n can be characterized again in several other ways. Notably, ' Differential equation : the function ϕ λ prq is the unique smooth solution to the differential equation `B B r ˘2ϕ λ `pn ´1qpcoth rq `B B r ˘ϕλ `pλ 2 `ρ2 q ϕ λ " 0 , which is normalized by ϕ λ p0q " 1. ' Relation with classical special functions :

ϕ λ prq " ϕ n´2 2 ,´1 2 λ prq " 2 F 1 `ρ`iλ 2 , ρ´iλ 2 ; n 2 ; ´sinh 2 r ˘,
where ϕ α,β λ are the Jacobi functions and 2 F 1 the Gauss hypergeometric function. ' Same functional equations as [START_REF] Amri | Singular integral operators in Dunkl setting[END_REF] and [START_REF] Anker | An elementary proof of the positivity of the intertwining operator in one-dimensional trigonometric Dunkl theory[END_REF]. ' The spherical functions are the matrix coefficients ϕ λ pxq "

Γp n 2 q 2 π n 2
xπ λ pxq1, 1y , of the spherical principal series representations of G on L 2 pS n´1 q. ' According to the Harish-Chandra formula ϕ λ pxq " ż K dk e pρ´iλq hpk xq , the function ϕ λ is a spherical average of horocyclic waves. Let us make this integral representation more explicit : Γ ℓ pλq e ´2ℓr , with Γ 0 " 1.

ϕ λ prq " Γp n 2 q 2 π n 2 ż S n´1 dv x cosh
The spherical Fourier transform (or Harish-Chandra transform) of radial functions on H n is defined by

Hf pλq " ż H n dx f pxq ϕ λ pxq " Γp n 2 q 2 π n 2 ż `8 0 dr psinh rq n´1 f prq ϕ λ prq (11)
and the inversion formula reads

f pxq " 2 n´3 π ´n 2 ´1 Γp n 2 q ż `8 0 dλ |cpλq| ´2 Hf pλq ϕ λ pxq . ( 12
)
Remark 2.7.

' The Plancherel density reads

|cpλq| ´2 " π 2 2n´4 Γp n 2 q 2 źn´3 2 j"0 pλ 2 `j 2 q
in odd dimension, and

|cpλq| ´2 " π 2 2n´4 Γp n 2 q 2 λ tanh πλ źn 2 ´1 j"0 " λ 2 `pj `1 2 q 2 ‰

Consider finally the transform

A ˚gpxq " ż K dk e ρ hpk xq gphpkxqq ,
which is dual to the Abel transform, i.e., ż

H n dx f pxq A ˚gpxq " ż `8
´8 dr Af prq gprq , and which is an isomorphism between C 8 even pRq and C 8 rad pH n q. It is given explicitly by

A ˚g prq " 2 n´1 2 Γp n 2 q ? π Γp n´1 2 q psinh rq 2´n ż r 0 ds pcosh r ´cosh sq n´3 2 gpsq
and its inverse by Applications. Spherical Fourier analysis is an efficient tool for solving invariant PDEs on H n . Here are some examples of evolution equations. ' The heat equation # B t upx, tq " ∆ x upx, tq upx, 0q " f pxq can be solved explicitly by means of the inverse Abel transform. Specifically, upx, tq " f ˚ht pxq , where the heat kernel is given by

pA ˚q´1 f prq " ? π 2 n´1 2 Γp n 2 q B B r `1 sinh r B sinh r
h t prq " 1 2 n`1 2 π n 2 t ´1 2 e ´ρ2 t `´1 sinh r B B r ˘n´1 2 e ´r2 4 t
in odd dimension and by

h t prq " p2πq ´n`1 2 t ´1 2 e ´ρ2 t ż `8 |r| ds ? cosh s ´cosh r `´B Bs ˘`´1 sinh s B Bs ˘n 2 ´1 e ´s2 4 t
in even dimension. Moreover, the following global estimate holds :

h t prq -e ´ρ2 t e ´ρ r e ´r2 4 t ˆ# t ´3 2 p1`rq if t 1`r, t ´n 2 p1`rq n´1 2 if 0 ă t 1`r. (13)
' Similarly for the Schrödinger equation # i B t upx, tq " ∆ x upx, tq, upx, 0q " f pxq.

In this case upx, tq " f ˚h´it pxq, where

h ´it prq " 1 2 n`1 2 π n 2 e i π 4 signptq |t| ´1 2 e ´i ρ 2 t `´1 sinh r B B r ˘n´1 2 e ´i r 2 4 t
in odd dimension,

h ´it prq " p2πq ´n`1 2 e i π 4 signptq |t| ´1 2 e ´i ρ 2 t ˆż `8 |r| ds ? cosh s ´cosh r `´B Bs ˘`´1 sinh s B Bs ˘n 2 ´1 e ´i s 2 4 t
in even dimension, and

|h ´it prq| À e ´ρ r ˆ# |t| ´3 2 p1`rq if |t| 1`r |t| ´n 2 p1`rq n´1 2 if 0 ă |t| 1`r in all dimensions. ' The shifted wave equation # B 2 t upx, tq " `∆x `ρ2 ˘upx, tq upx, 0q " f pxq, B t | t"0 upx, tq " gpxq
can be solved explicitly by means of the inverse dual Abel transform. Specifically, upt, xq " in even dimension.

1 2 n`1 2 π n´1 2 B B t `1 sinh t B B t ˘n´3 2 ! 1 sinh t ż Spx,|t|q dy f pyq ) `1 2 n`1 2 π n´1 2 `1 sinh t B B t ˘n´3 2 ! 1 sinh t ż Spx,

Spherical Fourier analysis on homogeneous trees.

A homogeneous tree is a connected graph, with no loop and with the same number of edges at each vertex. Let us denote by T q the set of vertices of the homogeneous tree with q `1 ą 2 edges. It is equipped with the counting measure and with the geodesic distance, given by the number of edges between two points. The volume of any sphere Spx, rq of radius r P N is given by δprq "

# 1 if r " 0, pq `1q q r´1 if r P N ˚.
Once we have chosen an origin 0 P T q and an oriented geodesic ω : Z Ñ T q through 0, let us denote by |x| P N the distance of a vertex x P T q to the origin and by hpxq P Z its horocyclic height (see Figure 7). Let G be the isometry group of T q and let K be the stabilizer of 0. Then G is a locally compact group, K is a compact open subgroup, and T q « G{K . Remark 2.9. If q is a prime number, then T q « PGLp2, Q q q{ PGLp2, Z q q. The combinatorial Laplacian ∆ on T q is defined by ∆f " Af ´f , where A denotes the average operator Af pxq "

1 q `1 ÿ yPSpx,1q
f pyq .

Remark 2.10. Notice that f is harmonic, i.e., ∆f " 0 if and only if f has the mean value property, i.e., f " Af .

The spherical functions ϕ λ on T q are the normalized radial eigenfunctions of ∆, or equivalently A. Specifically, # A ϕ λ " γpλq ϕ λ , ϕ λ p0q " 1 , where γpλq " q iλ `q´iλ q 1{2 `q´1{2 . Remark 2.11. The spherical functions on T q can be characterized again in several other ways. Notably,

ϕ λ prq " # cpλq q p´1{2`iλqr `cp´λq q p´1{2´iλqr if λ P C τ 2 Z, p´1q j r `1 `q1{2 ´q´1{2 q 1{2 `q´1{2 r ˘q´r{2 if λ " τ 2 j , ( 14
)
where cpλq "

1 q 1{2 `q´1{2 q 1{2 `i λ ´q´1{2´iλ q i λ ´q´iλ
and τ " 2π log q . Notice that ( 14) is even and τ -periodic in λ. ' Same functional equations as [START_REF] Amri | Singular integral operators in Dunkl setting[END_REF] and [START_REF] Anker | An elementary proof of the positivity of the intertwining operator in one-dimensional trigonometric Dunkl theory[END_REF]. ' The spherical functions are the bi-K-invariant matrix coefficients of the spherical principal series representations of G on L 2 pB T q q.

The spherical Fourier transform of radial functions on T q is defined by

Hf pλq " ÿ xPTq f pxq ϕ λ pxq " f p0q `q1{2 `q´1{2 q 1{2 ÿ `8 r"1 q r f prq ϕ λ prq (15)
and the inversion formula reads f prq "

q 1{2 q 1{2 `q´1{2 1 τ ż τ {2 0 dλ |cpλq| ´2 Hf pλq ϕ λ prq . ( 16
)
Consider the Abel transform

Af phq " q h 2 ÿ xPTq hpxq"h f p|x|q " q |h| 2 f p|h|q `q ´1 q ÿ `8 j"1 q |h| 2 `j f p|h|`2j q ,
which is essentially the horocyclic Radon transform restricted to radial functions. Then the following commutative diagram holds, let say in the Schwartz space setting :

C 8 even pR{τ Zq H Õ Ô F S rad pT q q ÝÑ A S even pZq
Here S even pZq denotes the space of even functions on Z such that sup rPN ˚r m |f prq| ă `8 @ m P N, S rad pT q q the space of radial functions on T q , whose radial part coincides with q ´r 2 SpNq, Ff pλq " ÿ hPZ q iλh f phq is a variant of the Fourier transform on Z, and each arrow is an isomorphism. The inverse Abel transform is given by A ´1f prq " ÿ `8 j"0 q ´r 2 ´j f pr `2jq ´f pr `2j `2q ( .

Consider finally the transform

A ˚g prq " 1 δprq ÿ xPTq |x|"r q hpxq 2 g `hpxq ˘,
which is dual to the Abel transform, i.e., ÿ

xPTq f pxq A ˚gp|x|q " ÿ hPZ Af phq gphq ,
and which is an isomorphism between the space of all even functions on Z and the space of all radial functions on T q . It is given explicitly by A ˚g p0q " 0, and

A ˚g prq " 2 q q `1 q ´r 2 f prq `q ´1 q `1 q ´r 2 ÿ ´r ă h ă r h has same parity as r f phq if r P N ˚. Its inverse is given by pA ˚q´1 f p0q " f p0q, pA ˚q´1 f phq " q 1{2 `q´1{2 2 q h´1 2 f phq ´q ´q´1 2 q ´h 2 ÿ 0ă r odd ă h q r f prq if h P N is odd, and
pA ˚q´1 f phq " q 1{2 `q´1{2 2 q h´1 2 f phq ´q1{2 ´q´1{2 2 q ´h´1 2 f p0q ´q ´q´1 2 q ´h 2 ÿ 0ă r even ă h q r f prq if h P N ˚is even.
Applications. Let us use spherical Fourier analysis to study discrete evolution equations on T q , as we did in the differential setting on H n .

' Consider the heat equation with discrete time t P N # upx, t `1q ´upx, tq " ∆ x upx, tq upx, 0q " f pxq or, equivalently, the simple random walk upx, tq " A t f pxq " f ˚ht pxq .

Its density h t pxq vanishes unless |x| t have the same parity. In this case,

h t pxq - 1`|x| p1`tq ? 1`t´|x| γ t 0 q ´|x| 2 e ´t ψ p 1 `|x| 1 `t q , ( 17
)
where ψpzq " 1`z 2 logp1`zq´1 ´z 2 logp1´zq and γ 0 " γp0q " 2 q 1{2 `q´1{2 ă 1 is the spectral radius of A.

' The shifted wave equation with discrete time t P Z # γ 0 ∆ Z t upx, tq " `∆Tq x `1´γ 0 ˘upx, tq upx, 0q " f pxq, tupx, 1q ´upx, ´1qu{2 " gpxq can be solved explicitly by using the inverse dual Abel transform. Specifically,

upx, tq " C t f pxq `St gpxq , where # C t " M |t| ´M|t|´2 2 S t " signptq M |t|´1 and M t f pxq " q ´t 2 ÿ dpy,xq t t´dpy,xq even f pyq if t 0, while M t " 0 if t ă 0.
2.6. Comments, references and further results.

' Our main reference for classical special functions is [START_REF]DLMF (Digital Library of Mathematical Functions[END_REF].

' The spherical Fourier analysis presented in this section takes place on homogeneous spaces G{K associated with Gelfand pairs pG, Kq.

' The sphere S n and the hyperbolic space H n are dual symmetric spaces. By letting their curvature tend to 0, the Euclidean space R n is obtained as a limit. At the level of spherical functions, the duality is reflected by the relation ϕ S n ℓ pcos θ 1 q " ϕ H n λ prq between ( 9) and ( 10), when we specialize λ " ˘i pρ `ℓ q (ℓ P N) and take r " ˘i θ 1 . And ( 6) is a limit of ( 10) and ( 9) :

ϕ R n λ prq " lim εÑ0 ϕ H n λ ε
pεrq " lim ℓÑ`8 ϕ S n ℓ pcos λr ℓ q . Homogeous trees may be considered as discrete analogs of hyperbolic spaces. This is for instance justified by the structural similarity between H 2 « PSLp2, Rq{ PSOp2q and T q « PGLp2, Q q q{ PGLp2, Z q q when q is a prime number. At the analytic level, an actual relation is provided by the meta-theory developed by Cherednik [START_REF] Cherednik | Double affine Hecke algebras[END_REF], which includes as limit cases spherical Fourier analysis on T q and on H n , as well as on R n or on S n . ' The material in Subsection 2.4 generalizes to the class of Riemannian symmetric spaces of noncompact type and of rank 1, which consist of all hyperbolic spaces

H n " H n pRq, H n pCq, H n pHq, H 2 pOq,
and further to the class of Damek-Ricci spaces. One obtains this way a group theoretical interpretation of Jacobi functions ϕ α,β λ for infinitely many discrete parameters

α " m 1 `m2 ´1 2 , β " m 2 ´1 2 .
Our main references are [START_REF] Faraut | Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques[END_REF], [START_REF] Koornwinder | Jacobi functions and analysis on noncompact semisimple Lie groups[END_REF], [START_REF] Anker | Spherical analysis on harmonic AN groups[END_REF], [START_REF] Rouvière | Espaces de Damek-Ricci, géométrie et analyse[END_REF] and [START_REF] Anker | The shifted wave equation on Damek-Ricci spaces and on homogeneous trees[END_REF]. ' Our main references for Subsection 2.5 are [START_REF] Figà-Talamanca | Harmonic analysis on free groups[END_REF], [START_REF] Cowling | An overview of harmonic analysis on the group of isometries of a homogeneous tree[END_REF], and [START_REF] Anker | The shifted wave equation on Damek-Ricci spaces and on homogeneous trees[END_REF]. Evolution equations (heat, Schrödinger, wave) with continuous time were also considered on homogeneous trees (see [START_REF] Setti | L p and operator norm estimates for the complex time heat operator on homogeneous trees[END_REF], [START_REF] Medolla | The wave equation on homogeneous trees[END_REF], [START_REF] Cowling | Estimates for functions of the Laplace operator on homogeneous trees[END_REF] and [START_REF] Eddine | Schrödinger equation on homogeneous trees[END_REF]). ' Spherical Fourier analysis generalizes to higher rank (see [START_REF] Helgason | Groups and geometric analysis (Integral geometry, invariant differential operators, and spherical functions[END_REF], [START_REF] Gangolli | Harmonic analysis of spherical functions on real reductive groups[END_REF] for Riemannian symmetric spaces and [START_REF] Macdonald | Spherical functions on a group of p-adic type[END_REF], [START_REF] Parkinson | Spherical harmonic analysis on affine buildings[END_REF], [START_REF] Mantero | Macdonald formula for spherical functions on affine buildings[END_REF] for affine buildings). Classification :

type constant curvature rank 1 general case Euclidean R n p¸K{K compact S n " SpR n`1 q SpF n q U{K non compact H n " H n pRq H n pFq G{K p-

adic homogeneous trees affine buildings

Notation : ˝gC is a complex semisimple Lie algebra, ˝g is a noncompact real form of g C and g " k ' p a Cartan decomposition of g, ˝u " k ' ip is the compact dual form of g, ˝a is a Cartan subspace of p, ˝g " a ' m ' `'αPR g α ˘is the root space decomposition of pg, aq, ˝R`i s a positive root subsystem and a `the corresponding positive Weyl chamber in a, ˝n " ' αPR `gα is the corresponding nilpotent Lie subalgebra, ˝mα " dim g α is the multiplicity of the root α, ˝ρ "

1 2 ÿ αPR `mα ,
˝GC is a complex Lie group with finite center and Lie algebra g C , ˝G, U, K and N are the Lie subgroups of G C corresponding to the Lie subalgebras g, u, k and n. Special functions : ˝Bessel functions on p :

ϕ p λ pxq " ż K dk e i xλ,pAd kqxy @ λ P p C , @ x P p C . ( 18
)
˝Spherical functions on G{K :

ϕ G λ pxq " ż K dk e x iλ`ρ, apk xq y @ λ P a C , @ x P G, ( 19
)
where apyq denotes the a-component of y P G in the Iwasawa decomposition G " Npexp aqK . ˝Spherical functions on affine buildings are classical Macdonald polynomials (i.e.

Hall-Littlewood polynomials for the type r A ). ' The global heat kernel estimate [START_REF] Anker | The heat kernel on symmetric spaces, in Lie groups and symmetric spaces[END_REF] generalizes as follows to G{K :

h t pxq -t ´n 2 ! ź αPR ``1 `xα, x `y˘`1 `t `xα, x `y˘mα `m2α 2
)

ˆe´|ρ| 2 t ´xρ,x `y ´|x `|2 4 t , (20)
where R `is the set of positive indivisible roots in R and x `denotes the a `component of x P G in the Cartan decomposition G " Kpexp a `qK (see [START_REF] Anker | The heat kernel on symmetric spaces, in Lie groups and symmetric spaces[END_REF] and the references therein). ' Random walks are harder to analyze on affine buildings. A global estimate similar to [START_REF] Ben Saïd | Laguerre semigroup and Dunkl operators[END_REF] and [START_REF] Cowling | An overview of harmonic analysis on the group of isometries of a homogeneous tree[END_REF] was established in [START_REF] Anker | Heat kernel and Green function estimates on affine buildings of type r A r[END_REF] for the simple random walk on affine buildings of type r A 2 . In general, the main asymptotics of random walks were obtained in [START_REF] Trojan | Heat kernel and Green function estimates on affine buildings[END_REF].

2.7. Epilogue. This section outlines spherical Fourier analysis around 1980 (except for the later applications to evolution equations). In the 1980s, Heckman and Opdam addressed the following problem (which goes back to Koornwinder for the root system BC 2 ) : for any root system R, construct a continuous family of special functions on a generalizing spherical functions on the corresponding symmetric spaces G{K , as Jacobi functions (or equivalently the Gauss hypergeometric function) generalize spherical functions on hyperbolic spaces. This problem was solved during the 1990s, mainly by Cherednik, Dunkl, Heckman, Macdonald, and Opdam, and has actually given rise to a large theory of special functions associated to root system, which is nowadays often referred to as Dunkl theory.

Remark 2.12. A different generalization of hypergeometric functions to Grassmannians was developed by Aomoto and Gelfand at the end of the 20th century.

Rational Dunkl theory

Rational Dunkl theory originates from the seminal paper [START_REF] Dunkl | Differential-difference operators associated to reflection groups[END_REF]. This theory of special functions in several variables encompasses ' Euclidean Fourier analysis (which corresponds to the multiplicity k " 0), ' classical Bessel functions in dimension 1, ' generalized Bessel functions associated with Riemannian symmetric spaces of Euclidean type (which correspond to a discrete set of multiplicities k).

In this subsection, we use [START_REF] Rösler | Dunkl operators (theory and applications)[END_REF] (or alternately [START_REF]Harmonic and stochastic analysis of Dunkl processes[END_REF], pp. 1-69) as our primary reference and quote only later works. Our notation goes as follows (see the appendix for more details) : ' a is a Euclidean vector space of dimension n, which we identify with its dual space, ' R is a root system, which is reduced but not necessarily crystallographic, ' W is the associated reflection group, ' a `is a positive Weyl chamber in a, a `its closure, R `the corresponding positive root subsystem, and S the subset of simple roots, ' a `denotes the closed cone generated by R `, which is the dual cone of a `, ' for every x P a, x `denotes the element of the orbit Wx which lies in a `, ' w 0 denotes the longest element in W, which interchanges a `and ´a`, respectively R `and R ´" ´R`, ' k is a multiplicity, which will remain implicit in most formulae and which will be assumed to be nonnegative after a while, ' γ " ÿ

for every ξ P a. Hence r D pq " r D p ˝r D q , for all symmetric pi.e. W-invariant q polynomials p, q P Ppaq W . ' Skew-adjointness : Assume that k 0. Then, for every ξ P a, ż a dx δpxq D ξ f pxq gpxq " ´ża dx δpxq f pxq D ξ gpxq .

Dunkl kernel.

Theorem 3.6. For generic multiplicities k and for every λ P a C , the system

# D ξ E λ " xλ, ξ y E λ @ ξ P a, E λ p0q " 1,
has a unique smooth solution on a, which is called the Dunkl kernel.

Remark 3.7. In this statement, generic means that k belongs to a dense open subset K reg of K, whose complement is a countable union of algebraic sets. The set K reg is known explicitly and it contains in particular t k PK | Re k 0 u.

Definition 3.8. The generalized Bessel function is the average

J λ pxq " 1 |W | ÿ wPW E λ pwxq " 1 |W | ÿ wPW E wλ pxq . ( 22
)
Remark 3.9.

' Mind the possible formal confusion between [START_REF] Deléaval | Dunkl kernel associated with dihedral groups[END_REF] @ α PR, then ϕ p λ pxq " J iλ pxq @ λ P a C , @ x P a.

In next proposition, we collect some properties of the Dunkl kernel.

) 24
In next theorem, we collect the main properties of the Dunkl transform.

Theorem 3.12. ' The Dunkl transform is an automorphism of the Schwartz space Spaq. ' We have $ ' ' ' & ' ' ' % HpD ξ f qpλq " i xξ, λy Hf pλq @ ξ P a, Hpxξ, . y f q " i D ξ Hf @ ξ P a, Hpwf qpλq " Hf pwλq @ w P W, H " f pt . q ‰ pλq " |t| ´n´2γ pHf qpt ´1λq @ t P R ˚.

' Inversion formula :

f pxq " c ´2 rat ż a dλ δpλq Hf pλq E iλ pxq , ( 25
)
where c rat "

ż a dx δpxq e ´|x| 2 2 (26)
is the so-called Mehta-Macdonald integral. ' Plancherel identity : The Dunkl transform extends to an isometric automorphism of L 2 pa, δpxqdxq, up to a positive constant. Specifically,

ż a dλ δpλq |Hf pλq| 2 " c 2 rat ż a dx δpxq |f pxq| 2 .
' Riemann-Lebesgue Lemma : The Dunkl transform maps L 1 pa, δpxqdxq into C 0 paq. ' Paley-Wiener Theorem : The Dunkl transform is an isomorphism between C 8 c paq and the Paley-Wiener space PWpa C q, which consists of all holomorphic functions h :

a C ÝÑ C such that D R ą 0, @ N P N, sup λPa C p1`|λ|q N e ´R |Im λ| |hpλq| ă `8 . ( 27
)
More precisely, the support of f P C 8 c paq is contained in the closed ball Bp0, Rq if and only if h " Hf satisfies [START_REF] Dunkl | Orthogonal polynomials of several variables[END_REF]. Remark 3.13. ' Notice that (24) and (25) are symmetric, as the Euclidean Fourier transform (1) and its inverse (2), or the Hankel transform (3) and its inverse (4). ' In the W-invariant case, E ˘iλ pxq is replaced by J ˘iλ pxq in (24) and (25). ' The Dunkl transform of a radial function is again a radial function. ' The following sharper version of the Paley-Wiener Theorem was proved in [START_REF] Amri | Three results in Dunkl theory[END_REF], as a consequence of the corresponding result in the trigonometric setting psee Theorem 4.10 q and thus under the assumption that R is crystallographic.

) 30
Notice the lack of Gaussian behavior when ´xy t, in particular when y " ´x tends to infinity faster then ? t. ' The Dunkl Laplacian is the infinitesimal generator of a Feller-Markov process on a, which has remarkable features pBrownian motion with jumpsq and which has drawn a lot of attention in the 2000s. We refer to [START_REF]Harmonic and stochastic analysis of Dunkl processes[END_REF] and [START_REF] Demni | Dunkl operators (an overview )[END_REF] for probabilistic aspects of Dunkl theory.

Problem 3.17. Prove in general heat kernel estimates similar to (30). ' Symmetries :

Apwf q " wpAf q and V pwgq " wpVgq @ w P W , A " f pt . q ‰ pyq " |t| ´2γ pAf qptyq and V " gpt . q ‰ pxq " pVgqptxq @ t P R ˚.

' For every x P a, there is a unique Borel probability measure µ x on a such that Vgpxq "

ż a dµ x pyq gpyq . ( 32
)
The support of µ x is contained in the convex hull of Wx. Moreover, if k ą 0, the support of µ x is W-invariant and contains Wx. ' A is an automorphism of the spaces C 8 c paq and Spaq, while V is an automorphism of C 8 paq, with |Vgpxq| max y PcopWxq |gpyq| @ x P a.

The following integral representations, which follow from ( 22), ( 31) and ( 32), generalize ( 6) and ( 18) in the present setting.

Corollary 3.19. For every λ P a C , we have

E λ pxq " ż a dµ
x pyq e xλ, y y and J λ pxq "

ż a dµ W x pyq Cosh λ pyq ,
where Cosh λ is defined in [START_REF] Demni | Dunkl operators (an overview )[END_REF] and

µ W x " 1 |W | ÿ wPW µ w x .
Remark 3.20.

' The first three items in Theorem 3.18 hold for all multiplicities k P K reg . ' The following symmetries hold : # dµ wx pwyq " dµ x pyq @ w P W , dµ tx ptyq " dµ x pyq @ t P R ˚.

' In [START_REF] De Jeu | Asymptotic analysis for the Dunkl kernel[END_REF], it is conjectured that the measure µ x is absolutely continuous with respect to the Lebesgue measure under the following two assumptions : ˝x is regular pwhich means that xα, xy ‰ 0, for every α P Rq, ˝a is spanned by the roots α with multiplicity k α ą 0. ' These conjectures hold in dimension 1 phence in the product caseq, where

dµ x pyq " Γpk `1 2 q ?
π Γpkq p|x|`signpxq yq px 2 ´y2 q k´1 1I p´|x|,`|x|q pyq dy if x ‰ 0, while µ 0 is the Dirac measure at the origin.

) 33
' The generalized translations are defined by

pτ y f qpxq " c ´2 rat ż a dλ δpλq Hf pλq E iλ pxq E iλ pyq " pτ x f qpyq . ( 34
)
The key objects here are the tempered distributions

f Þ ÝÑ x ν x,y , f y , ( 35
)
which are defined by [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF] and which enter the product formula

E λ pxq E λ pyq " x ν x,y , E λ y . ( 36
)
Remark 3.22. ' When k " 0, [START_REF] Gallardo | Propriété de Liouville et équation de Poisson pour le laplacien généralisé de Dunkl[END_REF] reduces to the usual convolution on a, (34) to pτ y f qpxq " f px `yq, and ν x,y " δ x`y . ' In the W-invariant case, (33) becomes pf ˚gqpxq " c ´2 rat ż a dλ δpλq Hf pλq Hgpλq J iλ pxq and (36)

J λ pxq J λ pyq " x ν W x,y , J λ y , ( 37
)
where ν W x,y " Γpk `1 4 q Γpk `3 4 q .

Then M 1 and M Õ ? 2 as k Õ `8.

In general there is a lack information about [START_REF] Gallardo | Newtonian potentials and subharmonic functions associated to the Dunkl-Laplace operator[END_REF] and the following facts are conjectured [START_REF] Rösler | A positive radial product formula for the Dunkl kernel[END_REF]. x,y is a measure, notice that it is normalized by ż a dν pW q x,y pzq " 1 . (e) Young's inequality : For all 1 p, q, r 8 satisfying 1 p `1 q ´1 r " 1, there exists a constant C 0 such that

}f ˚g } L r C }f } L p }g } L q . ( 39
)
Beside the trivial L 2 setting and the one-dimensional case (hence the product case), here are two more situations where Problems 3.25 and 3.26 have been solved.

' Radial case [START_REF] Rösler | A positive radial product formula for the Dunkl kernel[END_REF]. Translations of radial functions are positive. Specifically, for radial functions f pzq " 1 f p|z|q and nonzero y P a, we have

pτ y f qpxq " ż a dµ y |y| pzq 1 f `a|x| 2 `|y| 2 `2 xx, zy |y| ˘. ( 40
)
Hence [START_REF] Gangolli | Harmonic analysis of spherical functions on real reductive groups[END_REF] holds if f or g is radial.

' Symmetric space case. Assume that the multiplicity k corresponds to a Riemannian symmetric space of Euclidean type. Then ν W x,y is a positive measure and ( 39) holds for W-invariant functions. For the four infinite families of classical root systems, it can be actually deduced from an earlier integral formula of Selberg. In general, the Mehta-Macdonald formula was proved by Opdam, first for crystallographic root systems [START_REF] Opdam | Some applications of hypergeometric shift operators[END_REF] and next for all root systems [START_REF] Opdam | Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group[END_REF]. His proof was simplified by Etingof [START_REF] Etingof | A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups[END_REF], who removed in particular the computer-assisted calculations used in the last cases.

' We have not discussed the shift operators, which move the multiplicity k by integers and which have proven useful in the W-invariant setting (see [START_REF] Opdam | Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group[END_REF]).

' The following asymptotics hold for the Dunkl kernel, under the assumption that k 0 psee [START_REF] De Jeu | Asymptotic analysis for the Dunkl kernel[END_REF]q : there exists υ : W ÝÑ C such that, for every w P W and for every λ, x P a `, lim tÑ`8 pitq γ e ´it xλ,w xy E itλ pwxq " υpwq δpλq

´1 2 δpxq ´1 2 . ( 41
)
If w " I, we have υpIq " p2πq ´n 2 c, where c is defined by [START_REF] Dunkl | An intertwining operator for the group B 2[END_REF], and ( 41) holds more generally when r t " it tends to infinity in the half complex space t r t P C | Re r t 0u.

If w ‰ I, the Dunkl kernel is expected to have a different asymptotic behavior, when Re r t becomes positive. In dimension 1, we have indeed, for any

0 ă ε π 2 , lim r tÑ`8 |arg r t | π 2 ´ε r t k`1 e ´r tλx E r tλ p´xq " 2 k´1 k Γpk`1 2 q ? π λ ´k´1 x ´k´1 .
This discrepancy plays an important role in [START_REF] Anker | The Hardy space H 1 in the rational Dunkl setting[END_REF].

' In the fourth item of Theorem 3.18, the sharper results about the support of µ x when k ą 0 were obtained in [START_REF] Gallardo | Support properties of the intertwining and the mean value operators in Dunkl's analysis[END_REF].

' Specific information is available in the W-invariant setting for the root systems A n .

In this case, an integral recurrence formula over n was obtained in [START_REF] Amri | Note on Bessel functions of type A N´1[END_REF] and [START_REF] Sawyer | A Laplace-type representation of the generalized spherical functions associated to the root systems of type A[END_REF] for J λ , by taking rational limits of corresponding formulae in the trigonometric case (see the tenth item in Subsection 4.7). Moreover, an explicit expression of µ W x is deduced in [START_REF] Sawyer | A Laplace-type representation of the generalized spherical functions associated to the root systems of type A[END_REF]. As a consequence, the support of µ x is shown to be equal to the convex hull of Wx, when k ą 0.

' The asymmetric setting is harder. Beyond the one-dimensional case (and the product case), explicit expressions of the measure µ x are presently available in some twodimensional cases. For the root system A 2 , two closely related expressions were obtained, first in [START_REF] Dunkl | Intertwining operators associated to the group S 3[END_REF] and recently in [START_REF] Amri | On the integral representations for Dunkl kernels of type A 2[END_REF]. For the root system B 2 , a complicated formula was obtained in [START_REF] Dunkl | An intertwining operator for the group B 2[END_REF] and a simpler one recently in [START_REF] Amri | Laplace-type integral representations of the generalized Bessel function and of the Dunkl kernel of type B 2[END_REF]. The case of dihedral root systems I 2 pmq is currently investigated psee [START_REF] Deléaval | Dunkl kernel associated with dihedral groups[END_REF], [START_REF] Constales | Explicit formulas for the Dunkl dihedral kernel and the pκ,aqgeneralized Fourier kernel[END_REF] and the references thereinq.

' In Lemma 3.23, the sharper result in the crystallographic case was obtained in [START_REF] Amri | Three results in Dunkl theory[END_REF].

' An explicit product formula was obtained in [START_REF] Rösler | Bessel convolutions on matrix cones[END_REF] for generalized Bessel functions associated with root systems of type B and for three one-dimensional families of multiplicities (which are two-dimensional in this case). The method consists in computing a product formula in the symmetric space case, which corresponds to a discrete set of multiplicities k, and in extending it holomorphically in k. The resulting measure lives in a matrix cone, which projects continuously onto a `, and its image |W | ν W x,y is a probability measure if k 0.

' Potential theory in the rational Dunkl setting has been studied in [START_REF] Mejjaoli | On a mean value property associated with the Dunkl Laplacian operator and applications[END_REF], [START_REF] Gallardo | Propriété de Liouville et équation de Poisson pour le laplacien généralisé de Dunkl[END_REF], [START_REF] Maslouhi | Harmonic functions associated to Dunkl operators[END_REF], [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF], [START_REF] Gallardo | Newtonian potentials and subharmonic functions associated to the Dunkl-Laplace operator[END_REF], [START_REF] Gallardo | Support properties of the intertwining and the mean value operators in Dunkl's analysis[END_REF], [START_REF] Gallardo | Dunkl-Riesz potentials of Radon measures[END_REF], [START_REF] Graczyk | On the Green function and Poisson integrals of the Dunkl Laplacian[END_REF] (see also [START_REF] Rejeb | Fonctions harmoniques et sous-harmoniques associées à des systèmes de racines[END_REF] and the references therein).

' Many current works deal with generalizations of results in Euclidean harmonic analysis to the rational Dunkl setting. Among others, let us mention ˝ [START_REF] Thangavelu | Convolution operator and maximal function for the Dunkl transform[END_REF] about the Hardy-Littlewood and the Poisson maximal functions, ˝ [START_REF] Amri | Riesz transforms for Dunkl transform[END_REF] and [START_REF] Amri | Singular integral operators in Dunkl setting[END_REF] about singular integrals and Calderon-Zygmund theory, ˝ [START_REF] Anker | The Hardy space H 1 in the rational Dunkl setting[END_REF] and [START_REF] Dziubański | Riesz transforms characterizations of Hardy spaces H 1 for the rational Dunkl setting and multidimensional Bessel operators[END_REF] about the Hardy space H 1 .

' A further interesting deformation of Euclidean Fourier analysis, encompassing rational Dunkl theory and the Laguerre semigroup, was introduced and studied in [START_REF] Ben Saïd | Laguerre semigroup and Dunkl operators[END_REF].

Trigonometric Dunkl theory

Trigonometric Dunkl theory was developed in the symmetric case by Heckman and Opdam in the 1980s, and in the non-symmetric case by Opdam and Cherednik in the 1990s. This theory of special functions in several variables encompasses ' Euclidean Fourier analysis (which corresponds to the multiplicity k " 0), ' Jacobi functions in dimension 1, ' spherical functions associated with Riemannian symmetric spaces of noncompact type (which correspond to a discrete set of multiplicities k).

In this subsection, we use [START_REF] Opdam | Lecture notes on Dunkl operators for real and complex reflection groups[END_REF] as our primary reference and quote only later works. We resume the notation of Section 3, with some modifications : ' the root system R is now assumed to be crystallographic but not necessarily reduced, ' r R denotes the subsystem of non-multipliable roots, ' the reference density in the case k 0 is now δpxq " ź

for every ξ P a.

Notice that the counterpart of Remark 3.2 holds in the present setting. In next theorem, we collect properties of Cherednik operators. The main one is again commutativity, which leads to Cherednik operators D p , for every polynomial p P Ppaq, and to their symmetric parts r D p on W-invariant functions.

Theorem 4.2.

' For any fixed multiplicity k, the Cherednik operators (42) commute pairwise. ' The Cherednik operators map the following function spaces into themselves :

C " e P ‰ , Ppaq, C 8 paq, C 8 c paq, S 2 paq " `Cosh ρ ˘´1 Spaq, . . .
where C " e P ‰ denotes the algebra of polynomials in e λ pλ P P q and Cosh ρ is defined in [START_REF] Demni | Dunkl operators (an overview )[END_REF]. ' W-equivariance : For every w P W and ξ P a, we have `w ˝Dξ ˝w´1 ˘f pxq " D w ξ f pxq `ÿαPR `X wR ´kα xα, wξ y f pr α xq .

Hence r D pq " r D p ˝r D q for all symmetric polynomials p, q P Ppaq W . ' Adjointness : Assume that k 0. Then, for every ξ P a, ż ' The Heckman-Opdam Laplacian is given by , where D j , respectively B j denote the Cherednik operators, respectively the partial derivatives with respect to an orthonormal basis of a. ' In dimension 1, the Cherednik operator is given by Df pxq " `B Bx ˘f pxq `

∆f pxq " ÿ n j"1 D 2 j f pxq " differential part r ∆f pxq hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj ÿ n j"1 B 2 j f pxq `ÿαPR `kα coth xα, xy 2 B α f pxq `|ρ| 2 f pxq ´ÿαPR `kα
k 1 1´e ´x `2 k 2 1´e ´2 x ( f pxq´f p´xq ( ´ρ f pxq " `B Bx ˘f pxq ` k 1 2 coth x 2 `k2 coth x ( f pxq´f p´xq ( ´ρ f p´xq
and the Heckman-Opdam Laplacian by

∆f pxq " `B Bx ˘2f pxq ` k 1 coth x 2 `2 k 2 coth x ( `B Bx ˘f pxq `ρ2 f pxq ´! k 1 4 sinh 2 x 2 `k2 sinh 2 x ) f pxq´f p´xq ( ,
where ρ " k 1 2 `k2 . 4.2. Hypergeometric functions. Theorem 4.4. Assume that k 0. Then, for every λ P a C , the system

# D ξ G λ " xλ, ξ y G λ @ ξ P a, G λ p0q " 1.
has a unique smooth solution on a, which is called the Opdam hypergeometric function.

Definition 4.5. The Heckman-Opdam hypergeometric function is the average

F λ pxq " 1 |W | ÿ wPW G λ pwxq . ( 43
)
Remark 4.6. ' Conversely, G λ pxq can be recovered by applying to F λ pxq a linear differential operator in x whose coefficients are rational functions of λ. ' The expression G λ pxq extends to a holomorphic function of λ P a C , x P a `i U and k P V , where U is a W-invariant open neighborhood of 0 in a and V is a W-invariant open neighborhood of tk P K | k 0u. ' The Heckman-Opdam hypergeometric function [START_REF] Heckman | An elementary approach to the hypergeometric shift operators of Opdam[END_REF] is characterized by the system # r D p F λ " ppλq F λ @ p P Ppaq W , F λ p0q " 1.

' In dimension 1, the Heckman-Opdam hypergeometric function reduces to the Gauss hypergeometric function 2 F 1 or, equivalently, to the Jacobi functions ϕ α,β λ :

F λ pxq " 2 F 1 pρ `λ, ρ ´λ; k 1 `k2 `1 2 ; ´sinh 2 x 2 q " ϕ k 1 `k2 ´1 2 , k 2 ´1 2 i 2λ
p x 2 q , and the Opdam hypergeometric function to a combination of two such functions :

G λ pxq " ϕ k 1 `k2 ´1 2 , k 2 ´1 2 i 2λ p x 2 q `ρ `λ 2k 1 `2k 2 `1 psinh xq ϕ k 1 `k2 `1 2 , k 2 `1 2 i 2λ p x 2 q .
' When k " 0, G λ pxq reduces to the exponential e xλ, xy and F λ pxq to Cosh λ pxq. ' The functions G ´ρ and F ´ρ are equal to 1. ' Spherical functions ϕ G λ on Riemannian symmetric space G{K of noncompact type are Heckman-Opdam hypergeometric functions. Specifically, if # R is the root system of pg, aq,

m α " dim g α , set # R " 2 R, k 2α " 1 2 m α . Then ϕ G λ pexp xq " F i λ 2 p2xq .
We collect in the next two propositions asymptotics and estimates of the hypergeometric functions.

Proposition 4.7. The following Harish-Chandra type expansions hold :

F λ pxq " ÿ wPW cpwλq Φ wλ pxq , G λ pxq " 1 ś αP r R `px λ,α _ yq ´1 2 k α{2 ´kαq ÿ wPW cpwλq Ψ w,λ pxq .
Here

cpλq " c 0 ź αPR `Γp xλ,α _ y`1 2 k α{2 q Γpxλ,α _ y`1 2 k α{2 `kα q
, where c 0 is a positive constant such that cpρq " 1, and

Φ
) 45
where r δpλq "

c 2 0 |cpiλq| 2 ź αP r R `´i xλ,α _ y`1 2 k α{2 `kα ´ixλ,α _ y " ź αPR `Γp ixλ,α _ y`1 2 k α{2 `kα q Γpixλ,α _ y`1 2 k α{2 q Γp´ixλ,α _ y`1 2 k α{2 `kα`1q Γp´ixλ,α _ y`1 2 k α{2 `1q (46)
and c trig is a positive constant. and its inverse [START_REF] Helgason | Groups and geometric analysis (Integral geometry, invariant differential operators, and spherical functions[END_REF] to

f pxq " c ´2 trig ż a dλ c 2 0 |cpiλq| 2 Hf pλq F iλ pxq . ( 48
)
It is an isomorphism between S 2 paq W " `Cosh ρ ˘´1 Spaq W and Spaq W , which extends to an isometric isomorphism, up to a positive constant, between L 2 pa, δpxqdxq W and L 2 pa, |cpiλq| ´2 dλq W . ' Formulae (47) and (48) are not symmetric, as the spherical Fourier transform [START_REF] Anker | Spherical analysis on harmonic AN groups[END_REF] and its inverse [START_REF] Anker | The shifted wave equation on Damek-Ricci spaces and on homogeneous trees[END_REF] on hyperbolic spaces H n , or their counterparts (15) and ( 16) on homogeneous trees T q . The asymmetry is even greater between (44) and [START_REF] Helgason | Groups and geometric analysis (Integral geometry, invariant differential operators, and spherical functions[END_REF], where the density (46) is complex-valued. ' There is no straightforward Plancherel identity for the full Cherednik transform [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF].

Opdam has defined in [START_REF] Opdam | Harmonic analysis for certain representations of graded Hecke algebras[END_REF] a vector-valued transform leading to a Plancherel identity in the non-W-invariant case. 4.4. Rational limit. Rational Dunk theory (in the crystallographic case) is a suitable limit of trigonometric Dunk theory, as Hankel analysis on R n is a limit of spherical Fourier analysis on H n . More precisely, assume that the root system R is both crystallographic and reduced. Then, ' the Dunkl kernel is the following limit of Opdam hypergeometric functions : ' The distribution µ x is most likely a probability measure, as in the rational setting. ' This is true in dimension 1 phence in the product caseq, where

E λ pxq " lim εÑ0 G ε ´1λ pεxq , ' the
dµ x pyq " # dδ x pyq if x " 0 or if k 1 " k 2 " 0. µpx, yq dy otherwise.
As far as it is concerned, the density µpx, yq vanishes unless |y| ă |x|. In the generic case, where k 1 ą 0 and k 2 ą 0, it is given explicitly by

µpx, yq " 2 k 1 `k2 ´2 Γpk 1 `k2 `1 2 q ? π Γpk 1 q Γpk 2 q | sinh x 2 | ´2k 1 | sinh x| ´2k 2 ˆż |x| |y| dz psinh z 2 q pcosh z 2 ´cosh y 2 q k 1 ´1 pcosh x ´cosh zq k 2 ´1
ˆpsign xq e x 2 p2 cosh x 2 q ´e´y 2 p2 cosh z 2 q

( . (

) 50
In the limit case, where k 1 " 0 and k 2 ą 0,

µpx, yq " 2 k 2 ´1 Γpk 2 `1 2 q ? π Γpk 2 q
| sinh x| ´2k 2 pcosh x ´cosh yq k 2 ´1 psign xq pe x ´e´y q . (51)

In the other limit case, where k 1 ą 0 and k 2 " 0, the density is half of (51), with k 2 , x, y replaced respectively by k 1 , x 2 , y 2 .

4.6. Generalized translations, convolution and product formula. ' The generalized translations are defined by

pτ y f qpxq " c ´2 trig ż a dλ r δpλq Hf pλq G iλ pxq G iλ pyq . ( 52
)
The key objects are again the tempered distributions f Þ ÝÑ x ν x,y , f y defined by [START_REF] Krötz | Analysis on the crown domain[END_REF] and their averages ν W x,y "

' Assume that k 1 ą 0 and k 2 ą 0. Then νpx, y, zq " 2 k 1 ´2 Γpk 1 `k2 `1 2 q ? π Γpk 1 q Γpk 2 q signpxyzq | sinh x 2 sinh y 2 | ´2k 1 ´2k 2 pcosh z 2 q 2k 2 ˆż π 0 dχ psin χq 2k 2 ´1 ˆ" cosh x 2 cosh y 2 cosh z 2 cos χ ´1`cosh x `cosh y `cosh z 4 ‰ k 1 ´1 " sinh x `y `z 2 ´2 cosh x 2 cosh y 2 sinh z 2 `k1 `2k 2 k 2 cosh x 2 cosh y 2 cosh z 2 psin χq 2 `sinh z ´sinh x ´sinh y 2 cos χ ‰ . ' Assume that k 1 " 0 and k 2 ą 0. Then νpx, y, zq " 2 2k 2 ´1 Γpk 2 `1 2 q ? π Γpk 2 q signpxyzq |psinh xqpsinh yq| ´2k 2 ˆ" sinh x `y `z 2 sinh ´x `y `z 2 sinh x ´y `z 2 sinh x `y ´z 2 ‰ k 2 ˆ" sinh x `y ´z 2 ‰ ´1 e
x `y ´z 2 . ' In the other limit case, where k 1 ą 0 and k 2 " 0, the density is again half of the previous one, with k 2 , x, y replaced respectively by k 1 , x 2 , y 2 . In higher dimension, we have the trigonometric counterparts of Problems 3.25 & 3.26 but fewer results than in the rational case. In particular, there is no formula like [START_REF]Harmonic and stochastic analysis of Dunkl processes[END_REF] for radial functions. A new property is the following Kunze-Stein phenomenon, which is typical of the semisimple setting and which was proved in [START_REF] Anker | Opdam's hypergeometric functions pproduct formula and convolution structure in dimension 1q[END_REF] (see also [START_REF] Ayadi | Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique[END_REF]) and [START_REF] Trimèche | Hypergeometric convolution structure on L p -spaces and applications for the Heckman-Opdam theory[END_REF]. ' The hypergeometric functions x Þ ÝÑ G λ pxq and x Þ ÝÑ F λ pxq extend holomophically to a tube a `i U in a C . The optimal width for F λ was investigated in [START_REF] Krötz | Analysis on the crown domain[END_REF]. ' Proposition 4.7 was obtained in [START_REF] Opdam | Harmonic analysis for certain representations of graded Hecke algebras[END_REF]. The asymptotic behavior of F λ was fully determined in [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF]. This paper contains in particular a proof of the estimate

F λ pxq - ! ź αP r R xα, λy‰0 `1`xα, xy ˘) e xλ´ρ,xy @ λ, x P a `,
which was stated in [START_REF] Schapira | Contributions to the hypergeometric function theory of Heckman and Opdam (sharp estimates, Schwartz space, heat kernel )[END_REF] (see also [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines plaplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affinesq[END_REF]), and the following generalization of a celebrated result of Helgason & Johnson in the symmetric space case : F λ is bounded if and only if λ belongs to the convex hull of Wρ. ' The sharp estimates in Proposition 4.8 were obtained in [START_REF] Schapira | Contributions to the hypergeometric function theory of Heckman and Opdam (sharp estimates, Schwartz space, heat kernel )[END_REF] (see also [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines plaplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affinesq[END_REF]) and [START_REF] Rösler | Limit transition between hypergeometric functions of type BC and type A[END_REF].

' As in the rational case, we have not discussed the shift operators, which move the multiplicity k by integers and which have proven useful in the W-invariant setting (see [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Part I,Ch. 3], [START_REF] Opdam | Lecture notes on Dunkl operators for real and complex reflection groups[END_REF]Section 5]). ' Rational limits in Subsection 4.4 have a long prehistory. In the Dunkl setting, they have been used for instance in [START_REF] Rösler | Positivity of Dunkl's intertwining operator via the trigonometric setting[END_REF], [START_REF] Ben Saïd | Bessel functions for root systems via the trigonometric setting[END_REF], [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF], [START_REF] Amri | Three results in Dunkl theory[END_REF], [START_REF] Amri | Note on Bessel functions of type A N´1[END_REF], [START_REF] Sawyer | A Laplace-type representation of the generalized spherical functions associated to the root systems of type A[END_REF], . . . (seemingly first and independently in preprint versions of [START_REF] Ben Saïd | Bessel functions for root systems via the trigonometric setting[END_REF] and [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF]). ' There are other interesting limits between special functions occurring in Dunkl theory.

For instance, in [START_REF] Rösler | Limit transition between hypergeometric functions of type BC and type A[END_REF] and [START_REF] Rösler | Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type BC[END_REF], Heckman-Opdam hypergeometric functions associated with the root system A n´1 are obtained as limits of Heckman-Opdam hypergeometric functions associated with the root system BC n , when some multiplicities tend to infinity. See [START_REF] Rösler | A limit relation for Dunkl-Bessel functions of type A and B, Symmetry Integrability Geom[END_REF] for a similar result about generalized Bessel functions. ' The expressions [START_REF] De Jeu | Asymptotic analysis for the Dunkl kernel[END_REF] are substitutes for the integral representations [START_REF] Anker | The Hardy space H 1 in the rational Dunkl setting[END_REF] and [START_REF] Constales | Explicit formulas for the Dunkl dihedral kernel and the pκ,aqgeneralized Fourier kernel[END_REF]. A different integral representation of F λ is established in [START_REF] Sun | A new integral formula for HeckmanÐ-Opdam hypergeometric functions[END_REF]. ' Formula (50) was obtained in [START_REF] Anker | An elementary proof of the positivity of the intertwining operator in one-dimensional trigonometric Dunkl theory[END_REF] and used there to prove the positivity of µ x when k 1 ą 0 and k 2 ą 0. A more complicated expression was obtained previously in [START_REF] Gallardo | Positivity of the Jacobi-Cherednik intertwining operator and its dual[END_REF] and in [START_REF] Ayadi | Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique[END_REF]. It was used in [START_REF] Gallardo | Positivity of the Jacobi-Cherednik intertwining operator and its dual[END_REF] to disprove mistakenly the positivity of µ x . Later on, the positivity of µ x in the general case was investigated in [START_REF] Trimèche | Positivity of the transmutation operators and absolute continuity of their representing measures for a root system on R d[END_REF], using the positivity of the heat kernel. ' It is natural to look for recurrence formulae over n for the five families of classical crystallographic root systems A n , B n , C n , BC n , D n (see the appendix). In the case of A n , an integral recurrence formula for F λ (or for Jack polynomials) was discovered independently by several authors (see for instance [START_REF] Sawyer | Spherical functions on symmetric cones[END_REF], [START_REF] Okounkov | Shifted Jack polynomials, binomial formula, and applications[END_REF], [START_REF] Hallnäs | A recursive construction of joint eigenfunctions for the hyperbolic nonrelativistic Calogero-Moser Hamiltonians[END_REF]). An explicit expression of µ W x is deduced in [START_REF] Sawyer | Spherical functions on symmetric cones[END_REF] and [START_REF] Sawyer | A Laplace-type representation of the generalized spherical functions associated to the root systems of type A[END_REF], first for x P a `and next for any x P a `. In particular, if k ą 0, then µ W

x is a probability measure, whose support is equal to the convex hull of Wx and which is absolutely continuous with respect to the Lebesgue measure, except for x " 0 where µ W

x " δ 0 . ' As in the rational case (see the eighth item in Subsection 3.7), an explicit product formula was obtained in [START_REF] Rösler | Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC[END_REF] and [START_REF] Voit | Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC[END_REF] for Heckman-Opdam hypergeometric functions associated with root systems of type BC and for certain continuous families of multiplicities. ' Probabilistic aspects of trigonometric Dunkl theory were studied in [START_REF] Schapira | Contributions to the hypergeometric function theory of Heckman and Opdam (sharp estimates, Schwartz space, heat kernel )[END_REF] and [START_REF] Schapira | The Heckman-Opdam Markov processes[END_REF] (see also [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines plaplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affinesq[END_REF]). Regarding the heat kernel h t px, yq, the estimate (20) was shown to hold for h t px, 0q and some asymptotics were obtained for h t px, yq. But there is no trigonometric counterpart of the expression (29), neither precise information like (30) about the full behavior of h t px, yq. ' The bounded harmonic functions for the Heckman-Opdam Laplacian were determined in [START_REF] Schapira | Bounded harmonic functions for the Heckman-Opdam Laplacian[END_REF].

Appendix A. Root systems

In this appendix, we collect some information about root systems and reflection groups. More details can be found in classical textbooks such as [START_REF] Humphreys | Reflection groups and Coxeter groups[END_REF] or [START_REF] Kane | Reflection groups and invariant theory[END_REF].

Definition A.1. Let a « R n be a Euclidean space. ' A pcrystallographicq root system in a is a finite set R of nonzero vectors satisfying the following conditions : (a) for every α P R, the reflection r α pxq " x ´2

Figure 1 . 2 . 1 2. 1 .

 1211 Figure 1. Relation between various special function theories

  ' the dilations a r : y Þ ÝÑ e ´ry in Model 2, and the subgroup N « R n´1 consisting of horizontal translations n υ : y Þ ÝÑ y `υ (υ P R n´1 ) in Model 2. Then we have ' the Cartan decomposition G " KA `K , which corresponds to polar coordinates inModel 3,

˘n´3 2 psinhRemark 2 . 8 .

 228 rq n´2 f prq ( in odd dimension and by pA ˚q´1 f prq " cosh r ´cosh s psinh sq n´1 f psq in even dimension. Notice that the spherical function ϕ λ prq is the dual Abel transform of the cosine function cos λs.

0 sphereFigure 6 .

 06 Figure 6. Ball picture of T 3

Figure 7 . 3 '

 73 Figure 7. Upper half-space picture of T 3

  αPR `kα , ' δpxq " ź αPR `| xα, xy| 2kα is the reference density in the case k 0. 3.1. Dunkl operators. Definition 3.1. The rational Dunkl operators, which are often simply called Dunkl operators, are the differential-difference operators defined by D ξ f pxq " B ξ f pxq `ÿαPR `kα xα, ξ y xα, xy f pxq ´f pr α xq (

Remark 3. 2 .' 2 ż 1 0Theorem 3 . 3 .

 22133 ' Notice that Dunkl operators D ξ reduce to partial derivatives B ξ when k " 0. ' The choice of R `plays no role in Definition 3.1, as ÿ αPR `kα xα, ξ y xα, xy f pxq ´f pr α xq ( xy f pxq ´f pr α xq ( . Dividing by xα, xy produces no actual singularity in (21), as f pxq´f prα xq xα, xy " ´1 xα, xy ż 1 0 dt B B t f px ´t xα _ , xy αq " 2 |α| dt B α f px ´t xα _ , xy αq . Commutativity is a remarkable property of Dunkl operators. Fix a multiplicity k. Then D ξ ˝Dη " D η ˝Dξ @ ξ, η P a. This result leads to the notion of Dunkl operators D p , for every polynomial p P Ppaq, and of their symmetric part r D p on symmetric (i.e. W-invariant) functions. Example 3.4. ' The Dunkl Laplacian is given by ∆f pxq " xα,xy B α f pxq ´ÿαPR `kα |α| 2 xα, xy 2 f pxq ´f pr α xq ( loooooooooooooooooomoooooooooooooooooon difference part , where D j , respectively B j denote the Dunkl operators, respectively the partial derivatives with respect to an orthonormal basis of a. ' In dimension 1, the Dunkl operator is given by Df pxq " `B Bx ˘f pxq `k x f pxq´f p´xq ( and the Dunkl Laplacian by Lf pxq " `B Bx ˘2f pxq `2k x `B Bx ˘f pxq ´k x 2 f pxq´f p´xq ( . Here are some other properties of Dunkl operators. Proposition 3.5. ' The Dunkl operators map the following function spaces into themselves : Ppaq, C 8 paq, C 8 c paq, Spaq, . . . More precisely, the Dunkl operators D ξ , with ξ P a, are homogeneous operators of degree ´1 on polynomials. ' W-equivariance : For every w P W and p P Ppaq, we have w ˝Dp ˝w´1 " D w p .

3. 5 .dy

 5 Intertwining operator and (dual) Abel transform. Consider the Abel transform A " F ´1 ˝H , which is obtained by composing the Dunkl transform H with the inverse Euclidean Fourier transform F ´1 on a, and the dual Abel transform A ˚, which satisfies ż a dx δpxq f pxq A ˚gpxq " ż a Af pyq gpyq . Theorem 3.18. ' The dual Abel transform A ˚coincides with the intertwining operator V defined on polynomials by Dunkl and extended to smooth functions by Trimèche. ' Intertwining property : for every ξ P a, A ˝Dξ " B ξ ˝A and V ˝Bξ " D ξ ˝V . (31)

3. 6 .

 6 Generalized translations, convolution and product formula. Definition 3.21. ' The generalized convolution corresponds, via the Dunkl transform, to pointwise multiplication : pf ˚gqpxq " c ´2 rat ż a dλ δpλq Hf pλq Hgpλq E iλ pxq . (

Problem 3. 25 .

 25 (a) The distributions ν x,y are bounded signed Borel measures. (b) They are uniformly bounded in x and y. (c) The measures ν W x,y are positive. If ν pW q

Problem 3 .

 3 [START_REF] Dunkl | Intertwining operators associated to the group S 3[END_REF] and especially item (b) is important for harmonic analysis. It implies indeed the following facts, for the reference measure δpxqdx on a. Problem 3.26. (d) The generalized translations (34) are uniformly bounded on L 1 and hence on L p , for every 1 p 8.

3. 7 .

 7 Comments, references and further results. ' The computation of the integral (26) has a long history. A closed form was conjectured by Mehta for the root systems of type A and by Macdonald for general root systems.

' 1 .Definition 4 . 1 .

 141 Q denotes the root lattice, Q _ the coroot lattice and P the weight lattice, Cherednik operators. The trigonometric Dunkl operators, which are often called Cherednik operators, are the differential-difference operators defined by D ξ f pxq " B ξ f pxq `ÿαPR `kα xα, ξ y 1´e ´xα, xy f pxq ´f pr α xq ( ´xρ, ξ y f pxq

a

  dx δpxq D ξ f pxq gp´xq " ż a dx δpxq f pxq D ξ gp´xq .

|α| 2 4 sinh 2 xα, xy 2 f

 22 pxq ´f pr α xq ( looooooooooooooooooooooomooooooooooooooooooooooon difference part

Remark 4. 11 .

 11 ' In the W-invariant case, the Cherednik transform (44) reduces to Hf pλq " ż a dx δpxq f pxq F iλ p´xq (47)

Definition 4. 16 .'

 16 The generalized convolution corresponds, via the Cherednik transform, to pointwise multiplication : pf ˚gqpxq " c ´2 trig ż a dλ r δpλq Hf pλq Hgpλq G iλ pxq .

  r ´psinh rq v, e n y iλ´ρ "

	Remark 2.6. The asymptotic behavior of the spherical functions is given by the Harish-
	Chandra expansion
	where and	ϕ λ prq " cpλq Φ λ prq `cp´λq Φ ´λprq , Γpρq Γpiλq Γpiλ`ρq cpλq " Γp2ρq Φ λ prq " p2 cosh rq iλ´ρ 2 F 1 `ρ´iλ 2 , ρ`1´iλ 2 ; 1´iλ ; cosh ´2 r e piλ´ρqr ÿ `8 ℓ"0	"
		"	Γp n 2 q π Γp n´1 2 q n´1 ? 2 2 Γp n 2 q ? π Γp n´1 2 q	ż π 0 dθ psin θq n´2 " psinh rq 2´n 0 ds pcosh r ´cosh sq ż r cosh r ´psinh rqpcos θq 2 cos λs . n´3 ‰ iλ´ρ	(10)

  Here S rad pH n q denotes the L 2 radial Schwartz space on H n , which can be identified with pcosh rq ´ρ S even pRq, F the Euclidean Fourier transform on R and each arrow is an isomorphism. Thus the inversion of the spherical Fourier transform H boilds down to the inversion of the Abel transform A. In odd dimension,

	A ´1g prq " p2πq while, in even dimension, A ´1g prq " 1 2 n´1 2 π n 2 ż `8 |r| ? cosh s ´cosh r ´n´1 2 `´1 sinh r ds `´B B B r Bs ˘`´1 sinh s ˘n´1 2 gprq Bs B	˘n 2 ´1 gpsq .
	in even dimension. Notice the different behaviors
	at infinity, and at the origin.	|cpλq| ´2 " |cpλq| ´2 " 2 2n´4 Γp n π 2 q 2 |λ| n´1 π Γp n´1 2 q 2 2 2n´4 Γp n 2 q 2 λ 2
	' Observe that (11) and (12) are not symmetric, unlike (1) and (2), or (3) and (4).
	The spherical Fourier transform (11), which is somewhat abstract, can be bypassed by
	considering the Abel transform, which is essentially the horocyclic Radon transform
	restricted to radial functions. Specifically, ż
	Af prq " e ´ρr " p2πq n´1 N 2 Γp n´1 2 q Then the following commutative diagram holds, let say in the Schwartz space setting : dn f pna r q ż `8 ds sinh s pcosh s ´cosh rq n´3 2 f psq . |r|
		S even pRq H Õ Ô F S rad pH n q ÝÑ A S even pRq

  |t|q

												)
												dy gpyq
	in odd dimension and upt, xq "	2	1 n`1 2 π	n 2	B B|t| `B B pcosh tq	˘n 2	´1 ż	Bpx,|t|q	dy	?	f pyq cosh t ´cosh dpy,xq
	`1 2 n`1 2 π	n 2	signptq `B B pcosh tq	˘n 2	´1ż	Bpx,|t|q	dy

gpyq ? cosh t ´cosh dpy,xq

'

  and the classical Bessel function of the first kind J ν . ' Conversely, the Dunkl kernel E λ pxq can be recovered by applying to the generalized Bessel function J λ pxq a linear differential operator in x whose coefficients are rational functions of λ psee [62, proposition 6.8.(4)]q. ' In dimension 1, K reg is the complement of ´N ´1 2 in C. The generalized Bessel function (22) reduces to the modified Bessel function encountered in Subsection 2.2 : It can be also expressed in terms of the confluent hypergeometric function : E λ pxq " When k " 0, the Dunkl kernel E λ pxq reduces to the exponential e xλ, xy and the generalized Bessel function J λ pxq to

	Γpk `1 2 q ? π Γpkq " e λx Γp2k `1q ż `1 ´1 du p1´uq k´1 p1`uq k e λxu ż 1 Γpkq Γpk `1q 0 loooooooooooooooooooooomoooooooooooooooooooooon dv v k´1 p1´vq k e ´2λxv	.
	1 F 1 pk; 2k`1; ´2λxq
	Cosh λ pyq "	1 |W |	ÿ	wPW	e xwλ, y y .	(23)
	' As far as we know, the non-symmetric Dunkl kernel had not occurred previously in special functions, group theory or geometric analysis.

J λ pxq " j k´1 2 pλxq .

The Dunkl kernel is a combination of two such functions :

E λ pxq " j k´1 2 pλxq loooomoooon even `λx 2k`1 j k`1 2 pλxq looooooomooooooon odd ' Bessel functions

[START_REF] Cherednik | Double affine Hecke algebras[END_REF]

on Riemannian symmetric spaces of Euclidean type p ¸K{K are special cases of

[START_REF] Deléaval | Dunkl kernel associated with dihedral groups[END_REF]

, corresponding to crystallographic root systems and to certain discrete sets of mutiplicities. More precisely, if ˝g " p ' k is the associated semisimple Lie algebra, ˝a is a Cartan subspace of p, ˝R is the root system of pg, aq, ˝mα is the multiplicity of α P R, and ˝R is the subsystem of indivisible roots in R, ˝kα " mα`m 2 α 2

  Assume that k 0. Then, for every ξ 1 , . . . , ξ N P a,ˇˇB ξ 1 . . . B ξ N E λ pxq ˇˇ |ξ 1 | . . . |ξ N | |λ| N e xpReλq `, pRe xq `y @ λ P a C , @ x P a C .

	' Symmetries :	$ ' ' ' &	E
		' ' ' %

Proposition 3.10. ' Regularity : E λ pxq extends to a holomorphic function in λ P a C , x P a C and k PK reg . λ pxq " E x pλq, E wλ pwxq " E λ pxq @ w P W, E λ ptxq " E tλ pxq @ t P C, E λ pxq " E λ pxq when k 0.

' Positivity : Assume that k 0. Then, 0 ă E λ pxq e xλ `, x `y @ λ P a, @ x P a.

' Global estimate : 3.3. Dunkl transform. From now on, we assume that k 0. Definition 3.11. The Dunkl transform is defined by Hf pλq " ż a dx δpxq f pxq E ´iλ pxq . (

  Given a W-invariant convex compact neighborhood C of the origin in a, consider the gauge χpλq " max xPC xλ, xy. Then the support of f P C 8 c paq is contained in C if and only if its Dunkl transform h " Hf satisfies the condition @ N P N, sup λPa C p1`|λ|q N e ´χpIm λq |hpλq| ă `8 . In next proposition, we collect some properties of the heat kernel established by Rösler. s px, zq h t pz, yq . ' Expression by means of the Dunkl kernel : h t px, yq " c ´1 rat p2tq ´n 2 ´γ e ´1 rat p2tq ´n 2 ´γ max w PW e

	Problem 3.14. Extend the latter result to the non-crystallographic case. 3.4. Heat kernel. The heat equation # ' h t px, yq is an smooth symmetric probability density. More precisely, ˝ht px, yq is an analytic function in pt, x, yq P p0, `8qˆaˆa, ˝ht px, yq " h t py, xq, ˝ht px, yq ą 0 and ş a dy δpyq h t px, yq " 1. ' Semigroup property : h s`t px, yq " ż a dz δpzq h ´|x| 2 4 t ´|y| 2 4 t E x ? 2 t `y ? ´|wx´y | 2 4 t B Proposition 3.15. $ ' ' & ' ' % t ´k´1 2 e ´x2 `y2 4 t if |xy| t, t 1 2 p´xyq ´k´1 e ´px`yq 2 4 t if ´xy t. t ´1 2 pxyq ´k e ´px´yq 2 4 t if xy t,	(28) (

t upx, tq " ∆ x upx, tq upx, 0q " f pxq can be solved via the Dunkl transform (under suitable assumptions). This way, one obtains upx, tq " ż a dy δpyq f pyq h t px, yq , where the heat kernel is given by h t px, yq " c ´2 rat ż a dλ δpλq e ´t |λ| 2 E iλ pxq E ´iλ pyq @ t ą 0 , @ x, y P a . 2t ˘@ t ą 0, @ x, y P a .

(

29

)

' Upper estimate :

h t px, yq c . Remark 3.16.

' In

[START_REF] Anker | The Hardy space H 1 in the rational Dunkl setting[END_REF]

, the following sharp heat kernel estimates were obtained in dimension 1 pand also in the product caseq : h t px, yq -

  t|z| 2 ´p|x|´|y|q 2 u k´1 tp|x|`|y|q 2 ´|z| 2 u k´1 t2|x||y||z|u 2 k ´1if x, y, z P R ˚satisfy the triangular inequality ˇˇ|x|´|y| ˇˇă |z| ă |x|`|y| and νpx, y, zq " 0

	where
	νpx, y, zq "	Γpk `1 2 q ? π Γpkq	pz `x`yqpz `x´yqpz ´x`yq 2 xy z
	otherwise. Moreover, let M " sup x,y PR	ż	R	d|ν x,y |pzq "	? 2	rΓpk`1 2 qs 2
						1	ÿ
						|W |
			$ ' & ' %	νpx, y, zq |z| 2k dz if x, y P R dδ y pzq if x " 0 , dδ x pzq if y " 0 ,	˚,
							x	`y
							x	´y
	Figure 8. Picture of the set (38) for the root system B 2

wPW ν w x,w y . Lemma 3.23. The distributions (35) are compactly supported. ' Specifically, ν x,y is supported in the spherical shell z P a ˇˇˇˇ| x|´|y| ˇˇ |z| |x|`|y| ( .

' Assume that W is crystallographic. Then ν x,y is actually supported in z P a ˇˇz `ď x ``y `, z `ě y ``w 0 x `and x ``w 0 y `( ,

[START_REF] Gallardo | Positivity of the Jacobi-Cherednik intertwining operator and its dual[END_REF]

where ď denotes the partial order on a associated with the cone a `.

Example 3.24. In dimension 1, ν x,y is a bounded signed measure. Specifically, dν x,y pzq "

  All functions G λ with λ P a are strictly positive. ' The ground function G 0 has the following behavior : For every λ P a C , µ P a and x P a, we have |G λ`µ pxq| e xpRe λq `, x `y G µ pxq .In particular, the following estimates hold, for every λ P a C and x P a,|G λ pxq| G Reλ pxq G 0 pxq e xpRe λq `.4.3. Cherednik transform. From now on, we assume that k 0. Definition 4.9. The Cherednik transform is defined by The Cherednik transform is an isomorphism between C 8 c paq and the Paley-Wiener space PWpa C q. More precisely, let C be a W-invariant convex compact neighborhood of the origin in a and let χpλq " max xPC xλ, xy be the associated gauge. Then the support of f P C 8 c paq is contained in C if and only if h " Hf satisfies (28).

	ż
	λ pxq " are converging series, for generic λ P a C and for every x P a ÿ ℓPQ `Γℓ pλq e xλ´ρ´ℓ, xy , Ψ w,λ pxq " ÿ ℓPQ `Γℓ pw, λq e xwλ´ρ´ℓ, xy `. Proposition 4.8. Assume that k 0. ' G 0 pxq -! ź αP r R xα, xy 0 `1`xα, xy ˘) e ´xρ,x `y @ x P a. In particular, G 0 pxq -! ź αP r R ``1 `xα, xy ˘) e ´xρ,x `y if x P a `, while G 0 pxq -e ´xρ,x `y if x P ´a`. a dx δpxq f pxq G iλ p´xq . In next theorem, we collect the main properties of the Cherednik transform. Theorem 4.10. ' The Cherednik transform is an isomorphism between the L 2 Schwartz space S 2 paq " `Cosh ρ ˘´1 Spaq and the Euclidean Schwartz space Spaq. ' Hf pλq " f pxq " c ´2 trig ' Paley-Wiener Theorem : ' Inversion formula : ż	(44)

a dλ r δpλq Hf pλq G iλ pxq , (

  Dunkl transform H rat is a limit case of the Cherednik transform H trig : pH rat f qpλq " lim εÑ0 ε ´n´2γ H trig rf pε ´1 . qs ( pε ´1λq , ' likewise for the inversion formulae (25) and (45) : ´1 ˝H , which is obtained by composing the Cherednik transform H with the inverse Euclidean Fourier transform F ´1 on a, and the dual Abel transform A ˚, which satisfies ż Remark 4.12. In [85], A ˚" V is called the trigonometric Dunkl intertwining operator and A " V ˚the dual operator. D ξ ˝V .' For every x P a, there is a unique tempered distribution µ x on a such that Vgpxq " x µ x , g y . Moreover, the support of µ x is contained in the convex hull of Wx. Corollary 4.14. For every λ P a C , we have G λ pxq " x µ x , e λ y and F λ pxq " x µ W

	pH ´1 rat f qpxq " const. lim εÑ0 ε n`2γ H ´1 trig rf pε . qs	(	pεxq .
	4.5. Intertwining operator and (dual) Abel transform. In the trigonometric set-
	ting, consider again the Abel transform
		A " F a dx δpxq f pxq A ˚gpxq "	ż	a	dy Af pyq gpyq .
	Proposition 4.13.
	' For every ξ P a,	A ˝Dξ " B ξ ˝A	and V ˝Bξ " 1 |W | ÿ wPW µ w x .
	Remark 4.15.

x , Cosh λ y ,

[START_REF] De Jeu | Asymptotic analysis for the Dunkl kernel[END_REF]

where Cosh λ is defined in

[START_REF] Demni | Dunkl operators (an overview )[END_REF]

and µ W

x "

  Proposition 4.18. Let 1 p ă 2. Then there exists a constant C ą 0 such that }f ˚g} L 2 C }f } L p }g} L 2 , for every f P L p pa, δpxqdxq and g P L 2 pa, δpxqdxq. 4.7. Comments, references and further results. ' The joint action of the Cherednik operators D p , with p P Ppaq, and of the Weyl group W may look intricate. It corresponds actually to a faithful representation of a graded affine Hecke algebra [63]. ' Heckman [43] considered initially the following trigonometric version

	1 D ξ f pxq " B ξ f pxq	`ÿαPR `kα
	These operators are W-equivariant :
	and skew-invariant : ż	w ˝1D ξ	˝w´1 " 1 D w ξ ,
	α,βPR `kα k β 4

2 xα, ξ y coth xα, xy 2 f pxq ´f pr α xq ( of rational Dunkl operators, which are closely connected to (42) : D ξ f pxq " 1 D ξ f pxq ´ÿαPR `kα 2 xα, ξ y f pr α xq . a dx δpxq p 1 D ξ f qpxq gpxq " ´ża dx δpxq f pxq p 1 D ξ gqpxq , but they don't commute : r 1 D ξ , 1 D η s f pxq " ÿ xα, ξ y xβ , ηy ´xβ , ξ y xα, ηy ( f pr α r β xq .

  xα, xy |α| 2 α maps R onto itself, (b) 2 xα, β y |α| 2 P Z for all α, β P R. ' A root system R is reducible if it can be splitted into two orthogonal root systems, and irreducible otherwise. Remark A.2. ' Unless specified, we shall assume that R spans a. ' α _ " 2 |α| 2 α denotes the coroot corresponding to a root α. If R is a root system, then R _ is again a root system. ' Most root systems are reduced, which means that (c) the roots proportional to any root α are reduced to ˘α. Otherwise the only possible alignment of roots is ´2α, ´α, `α, `2α . A root α is called ˝indivisible if α 2 is not a root, ˝non-multipliable if 2α is not a root. ' We shall also consider non-crytallographic reduced root systems R, which satisfy (a) and (c), but not necessarily (b). Definition A.3. ' The connected components of t x P a | xα, xy ‰ 0 @ α P R u are called Weyl chambers. We choose any of them, which is called positive and denoted by a `. R `denotes the set of roots which are positive on a `. ' The Weyl or Coxeter group W associated with R is the finite subgroup of the orthogonal group Opaq generated by the root reflections t r α | α P R u. Remark A.4. ' The group W acts simply transitively on the set of Weyl chambers. ' The longest element w 0 in W interchanges a `and ´a`. ' B n (n 2) :a " R n R " t ˘ei | 1 i n u Y t ˘ei ˘ej | 1 i ă j n u a `" t x P R n | x 1 ą . . . ą x n ą 0 u W " t˘1u n ¸Sn ' C n (n 2) : a " R n R " t ˘2 e i | 1 i n u Y t ˘ei ˘ej | 1 i ă j n u a `" t x P R n | x 1 ą . . . ą x n ą 0 u W " t˘1u n ¸Sn ' BC n (n 1) : a " R n R " t ˘ei , ˘2 e i | 1 i n u Y t ˘ei ˘ej | 1 i ă j n u a `" t x P R ' I 2 pmq (m 3) : a " C R " t e i π j m | 0 j ă 2 m u a `" t z P C ˚| `1 2 ´1 m ˘π ă arg z ă π 2 u W " pZ{m Zq¸pZ{2Zq (dihedral group)The full list of irreducible root systems (crystallographic or reduced) includes in addition a finite number of exceptional cases :E 6 , E 7 , E 8 , F 4 , G 2 , H 3 , H 4 .Remark A.5. In the list above, ' the non crystallographic root systems are H 3 , H 4 and I 2 pmq with # m " 5, m 7, ' all root systems are reduced, with the exception of BC n , ' there are some redundancies in low dimension :A 1 ˆA1 « D 2 « I 2 p2q B 2 « C 2 « I 2 p4qpup to the root lengthqA 2 « I 2 p3q G 2 « I 2 p6qpup to the root lengthqThe 2-dimensional root systems (crystallographic or reduced) are depicted in Figure9.Remark A.7. ' In Dunkl theory, one assumes most of the time that k 0. ' Assume that R is crystallographic and irreducible. Then two roots belong to the same W-orbit if and only if they have the same length. Thus k takes at most three values. In the non crystallographic case, there are one or two W-orbits in R. Specifically, by resuming the classification of root systems, k takes ˝1 value in the following cases : A n , D n , E 6 , E 7 , E 8 , H 3 , H 4 , I 2 pmq with m odd, ˝2 values in the following cases : B n , C n , F 4 , G 2 , I 2 pmq with m even, ˝3 values in the case of BC n .

	A 1 ˆA1	BC 1 ˆBC 1
	$ ' ' ' &
	' ' ' %
	A 2	B 2
	C 2	BC 2
	G 2	I 2 p5q
	Figure 9. 2-dimensional root systems

' Every x P a belongs to the W-orbit of a single x `P a `.

There are six classical families of irreducible root systems :

' A n (n 1) : a " t x P R n`1 | x 0 `x1 `. . . `xn " 0 u R " t e i ´ej | 0 i ‰ j n u a `" t x P a | x 0 ą x 1 ą . . . ą x n u W " S n`1 n | x 1 ą . . . ą x n ą 0 u W " t˘1u n ¸Sn ' D n (n 3) : a " R n R " t ˘ei ˘ej | 1 i ă j n u a `" t x P R n | x 1 ą . . . ą |x n | u W " t ε P t˘1u n | ε 1 . . . ε n " 1 u¸S n Definition A.6.

A multiplicity is a W-invariant function k : R ÝÑ C.

Algébriques, www.fdpoisson.fr/madaca). We thank the organizers of the school and conference AAGADE 2015


Kamani Perera
email: kamaniperera2015@yahoo.com

Perera Kamani

Librarian
email: librarian@rcss.org94775965252

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

ETDs have changed traditional scenario of academic library settings. It has modified studying in education and research, but in spite of its great potential, ETDs have however to beat traditional paper based theses environment in the market. Universities, Research and Educational Institutes, Academic Libraries etc. must respond to growing presence of ETDs and ensure that users remain aware of ETD services offered by the organizations. Although, there are limitations to ETD's implementation, its application in academic sector is likely to grow over time. It is evident that ETDs can play a significant role in repositories and it may lead to a result-oriented approach by way of improving data skills of research scholars. It is, therefore, extremely important to provide accurate data sets to fulfill the information gap and at the same time supporting to accomplish their novel research by providing required ETDs in time.

Problem Statement

ETD is a collaborative effort of universities/organizations to promote creating, archiving, distributing and accessing metadata. With the emergence of new information technologies a significant amount of ETD repositories became available all over the world. Research from various perspectives is going on to evaluate them. Though ETDs are playing major role in institutional repositories in academic sector, there is still a gap of awareness and accessing relevant PhD thesis for student's research purposes. This study is focused to evaluate PhD students' awareness, frequency and expectations of ETD repository at RCSS library and to eliminate those gaps, if any, to facilitate in providing effective ETDs to research scholars.

Purpose of the Study

The purpose of present research was to discover the usage of ETDs by PhD students' who visit the RCSS library. In this research survey, the researcher has examined motivational factors that influence use of the ETDs. It provided an overview to the most frequently used ETDs in RCSS library repository. Such periodical evaluation helps the organization in ensuring the standard, performance, reliability, appearance, and commitment to delivery time of the required information.

Methodology --Data Collection

For this research study, questionnaire was distributed among the PhD students who have registered in the RCSS library in 2014--2015. Participants have been selected through an open invitation and prior consent has taken on individual basis before sending the questionnaire. The purpose of this procedure was to evaluate as to what extent PhD students utilize the ETDs offered by the RCSS.

Literature Review

ETDs are playing dynamic role in academic sector by adding value to the student's curriculum vitae while boosting their research knowledge in greater scale. This has been clearly indicated by [START_REF] Alemneh | Guidance Documents for Lifecycle Management of ETDs[END_REF] and according to them ETDs are generally support students' candidature for academic degrees, enriching CVs and adding value to their professional qualifications. Thus, it is very important to maintain and promote ETDs in academic libraries where students' carry out their research and education. [START_REF] Perera | ETDs for Research and Development: A Case Study from Regional Centre for Strategic Studies (RCSS) Sri Lanka. In: Evolving Genre of Electronic Theses and Dissertations for Knowledge Discover[END_REF] emphasized that the library of RCSS has been used open access websites, databases to build up its ETD repository and preserved it as PDF documents according to the respective subject fields. Most of the ETDs in RCSS repository are born digital and few of them are converted to digital. Similarly, [START_REF] Copeland | Electronic Theses and Dissertations: Promoting 'hidden[END_REF] pointed out that many universities are taking efforts to promote 'open access' to develop their institutional repositories, which contain the full text of ETDs, journal articles, book chapters, conference proceedings, reports etc.

To produce scholarly research piece, it is essential to think critically and engage with existing research data according to the researcher's particular subject area. At this juncture, ETDs have become major component, which produce novel research data to the scholarly field. [START_REF] Fox | Electronic Theses and Dissertation: A Source Book for Educators, Students and Librarians[END_REF] stated that the purpose of the ETD is to prepare the next generation of scholars to function effectively as knowledge workers in the information age. Similarly, [START_REF] Gaur | Evolving Genre of Electronic Theses and Dissertations for Knowledge Discover[END_REF] described the importance of ETDs in an academic sector. According to them, ETDs facilitate following activities for academic scholars.

--Topic identification --Comprehensive review --Literature review --Currency --Methodology/Theory Development --Future research --Test and measurement instruments --Unique topic area exploration --Writing styles --Formatting ideas [START_REF] Copeland | The development and promotion of electronic theses and dissertations (ETDs) within the UK[END_REF] also highlighted the benefits of ETD repositories for institutions and libraries. According to them that institutions benefit from improved access to the research findings of their staff and students. Increased levels of use of the material produced at a particular university can help to promote the research profile of the institution. In the same vein, libraries also benefit when theses and dissertations are made available in electronic format. It saves storage space, saves staff time spent retrieving and re--shelving hard--copy theses, reduces number of inter--library loan requests, easy collection of usage statistics and improve the customer satisfaction. [START_REF] Vijayakumar | Importance of Doctoral Theses and Its Access: A Literature Analysis[END_REF] pointed out that doctoral (PhD) theses are contribution to the world of knowledge. Further the results, conclusions and recommendations invariably have impact towards the economical and social up--liftment. In addition, the thought content also paves way to new vistas of investigation and encourage inter--disciplinary approach, which will strengthen the fabric of life styles in the society.

Data Analysis

The research survey in 2015 has proved that the scholarly community of the RCSS has used ETD collection mainly for research and development, and educational purposes (Figure 1).

Most of the users are coming under faculty members and research scholars whom they have used ETDs for research and development while others are postgraduates and undergraduates who used ETDs for educational purposes [START_REF] Perera | Scholarly Communication through e--Theses in Academic Libraries: A Case Study from Regional Centre for Strategic Studies (RCSS) Library, Sri Lanka[END_REF]. Figure 1. Purpose of using e--theses at RCSS

Goals

The objective of this research study was to evaluate the satisfaction level of PhD students' who are using ETDs provided by the RCSS Library. In this research survey it is found that the researchers are motivated to use the ETD repository due to their subject relevancy, reliability, accessibility and currency (Figure 2). This evaluation not only inspires more energetic ETD development environment which ultimately leads to an effective way of universal access but also endeavors for the systematic development of ETDs. Once PhD students' are familiar with the ETD technologies they tend to become critical thinkers of their own research study and will be able to produce more fruitful scholarly piece. The series of workshops are being conducted by the Library of RCSS for the purpose of improving writing skills for research and dissertations among graduate and postgraduate students. The students of the workshops are encouraged to make use the ETD repository at RCSS Library before they submit their research proposals to the respective universities. Plagiarized content can be easily discovered in electronic way. Therefore, the students are advised to avoid plagiarism by way of paraphrasing, summarizing and citing the original work. In this context, it is very important to cite the original work of paraphrased or summarized contents. Otherwise, it has considered as a plagiarized content. And also students are encouraged to discuss their own experience, challenges, successes and failures of their research study rather than borrowing other researchers' opinions to maintain the originality of their research study. [START_REF] Schopfel | Open Access to Research Data in Electronic Theses and Dissertations: An Overview[END_REF] pointed out that theses and dissertations constitute the significant part of the academic grey literature produced by the universities all over the world. Similarly, [START_REF] Goodfellow | Nurse Scholars Knowledge and Use of Electronic Theses and Dissertations[END_REF] stated that ETDs are the research outcomes of the research degree and postgraduate degree programmes. Due to the easy access of electronic format to wider audience, universities now a day have encouraged their research scholars and postgraduate students to submit their thesis and dissertations in electronic format. Suber (2012) also stated that theses and dissertations are the most useful kind of invisible scholarship and the most invisible kinds of useful scholarship. In the same vein, [START_REF] Cayabyab | A Review of Emerging ETD Initiatives, Challenges, and Future Developments[END_REF] mentioned that easy submission, access, dissemination and effective cost management are the factors that make the development of ETD repositories a success stories worldwide. Similarly, Mini Devi, Gana and Sri Ram (2015) also highlighted the importance of creation of ETD repositories in libraries and information Centres. They have emphasized that usage of ETDs help to uplift the new generation in exploring the unrevealed areas of research. It eventually supports PhD scholars to produce successful research output. Lipincott and Lynch (2010) also suggested that libraries could play a broader role in graduate education by arranging for the institutional repository services, which will enhance students 'dissertation experiences'. The RCSS has already taken steps to add ETDs to its e--library since 2009. PhD students are encouraged to access the ETDs at RCSS library and to study the research topics, problem statements, research questions, objectives and methodology etc. and how to produce good research proposal avoiding plagiarism. UNESCO (2015) also pointed out that ETDs are part of the scholarly communication compendium and a critical component in any of the research life cycle. Similarly, [START_REF] Fox | Overview of a Guide for Electronic Theses and Dissertations[END_REF] stated that access to ETD is viewed as the only resource open for those in developing countries who cannot afford to subscribe commercial database vendors; or attend international scholarly symposium and participate in inter--library (ILL) projects. It is crystal clear that ETDs are playing a dynamic role in academic libraries especially in developing countries where that get budgetary constraints to subscribe necessary databases. In this manner, scholars in the developing countries are immensely benefited from the open access ETD repositories in academic libraries. According to [START_REF] Fineman | Electronic Theses and Dissertations[END_REF] ETDs support highly expressive multimedia supplements, cost effective in comparison with print theses and dissertations, activities teach students about electronic publishing and information technology thus helping them to function more effective in the information age, promote easy access to current research and improved visibility for authors. [START_REF] Fineman | Electronic Theses and Dissertations in Music[END_REF] further stated that there are two types of ETDs such as 1. Convert to digital and 2. Born digital. The former one is generally digitized by the library and archives staff and to enhance the access to paper based theses and dissertations (TDs) and latter part is created by the students in the course of their research. In Sri Lanka, students generate their TDs in electronic format for supervision stage and final stage they should submit hard copies of their TDs to the university.

Conclusion

As there are growing programmes of postgraduate studies, it is very essential to maintain ETD repositories in academic libraries. Even in Sri Lanka, there are plenty of educational institutes who conduct such programmes other than the universities. Therefore, it is important to make awareness of ETD repositories by way of conducting outreach programmes, workshops etc. to provide the effective services to PhD scholars. Thus, RCSS has taken steps to conduct series of workshops on 'Skills Development for Research/Dissertation'. Postgraduates are immensely benefitted from these workshops and they are encouraged to make use ETD repository at RCSS Library. Students can investigate the existing research topics and will be able to select suitable research topic by avoiding repetition of research. On the other hand they can conduct further research on particular subject area to eliminate the gaps of that research. In this way ETDs provide proper guidance and open new window for a novel research findings. [START_REF] Gaur | Evolving Genre of Electronic Theses and Dissertations for Knowledge Discover[END_REF] pointed out that ETDs facilitate to unleash author's creativity and setting them on a path of inventiveness that can follow them throughout the career. Similarly, Ahmed, Alreyaee and Rahman (2014) stated that ETDs play a key role in highlighting the research impact and enhancing the rank of institutions. In the same vein, they emphasized that ETDs have become a major component of institutional repositories across the continent of Asia. In RCSS Library, ETDs play a dynamic role by adding value to the existing electronic repository. [START_REF] Moxley | American Universities should require Electronic Theses and Dissertations[END_REF] stated that in future, a university's quality would be linked to its digital library of theses and dissertations. In this research study, the RCSS has proved that how the Library become a 'Research Gem' due to the scholarly visibility of its ETD repository among similar academic libraries in the region.

Figure 2 .

 2 Figure 2. Motivational factors of using ETDs at RCSS