S Jaffard

Pointwise and directional regularity of nonharmonic Fourier series

Keywords:

We investigate how the regularity of nonharmonic Fourier series is related to the spacing of their frequencies. This is obtained by using a transform which simultaneously captures the advantages of the Gabor and Wavelet transforms. Applications to the everywhere irregularity of solutions of some PDEs are given. We extend these results to the anisotropic setting in order to derive directional irregularity criteria.

This paper is dedicated to the memory of Jean Morlet, who made me vividly aware of the importance of cross-fertilizations between mathematics and signal processing.

Jean Morlet always advocated the importance of comparing the possibilities of two key tools in signal processing: The wavelet transform and the Gabor transform. One purpose of the present paper is to show on a explicit problem that these tools should not be opposed, but can be combined in order to yield optimal results in the study of the regularity of nonharmonic Fourier series.

Let (λ n ) n∈IN be a sequence of points in IR d ; a function which can be written

f (x)= n∈IN a n e iλn•x , (1)
(where (a n ) n∈IN is a sequence of complex numbers) is called a nonharmonic Fourier series; note that the word "nonharmonic" points to the fact that we do not make the assumption that the frequencies λ n are a subset of a discrete subgroup of IR d ; see [START_REF] Young | A introduction to nonharmonic Fourier series[END_REF] for for a comprehensive study of the properties of nonharmonic Fourier series. Our purpose is to study the relationship between the regularity of such series and the spacing of their frequencies.

The first example of a Fourier series displaying large gaps between its frequencies was the function

R(t)= ∞ n=1 sin(n 2 t) n 2 , (2)
which Riemann expected to be a possible example of a continuous nowhere differentiable function, see [START_REF] Jaffard | The spectrum of singularities of Riemann's function[END_REF][START_REF] Luther | The differentiability of Fourier gap series and "Riemann's example" of a continuous nondifferentiable function[END_REF]. The relationship between the spacings of the frequencies of a Fourier series and its everywhere irregularity has been the subject of many investigations, starting with the famous example of the Weierstrass functions; in Section 1, we review these results and show how they can all be derived from a simple property of the Gabor-Wavelet transform: Theorem 7 yields general irregularity results for multidimensional nonharmonic Fourier series. In Section 2, we show that, surprisingly, this simple result is optimal only for Hölder exponents less than 1, and Theorem 11 shows how it can be improved for larger Hölder exponents. Extensions are worked out in Section 3, and applications to everywhere irregularity of solutions of PDEs are derived in Section 4. In Section 5, the notion of directional irregularity is discussed, and a new definition is proposed; the techniques of Section 1 are adapted in a non-isotropic framework, and we derive directional irregularity results for Fourier series whose frequencies have lacunarities in certain directions.

Pointwise T p u irregularity of lacunary Fourier series

The notion of pointwise regularity most commonly used in statements concerning lacunary Fourier series is the pointwise Hölder regularity. The natural setting for this notion is supplied by functions with slow growth, i.e. locally bounded functions f which satisfy ∃C, A > 0, ∀R, sup

B(0,R) |f (x)|≤C(1 + R) A ,
where B(x, R) denote the open ball centered at x and of radius R.

Definition 1 Let f :I R d → C be a function with slow growth and let α ≥ 0; f ∈ C α (x 0 ) if there exist R>0, C>0, and a polynomial P of degree less than α such that, for R small enough,

if |x -x 0 |≤R, then |f (x) -P (x -x 0 )|≤C|x -x 0 | α . ( 3
)
The Hölder exponent of f at x 0 is h f (x 0 ) = sup{α : f ∈ C α (x 0 )}.

This notion is only pertinent when applied to locally bounded functions; indeed, [START_REF] Caldéron | Local properties of solutions of elliptic partial differential equations[END_REF] implies that f is bounded in a neighbourhood of x 0 . We will derive pointwise irregularity results for solutions of PDEs where the natural function space setting is L p or a Sobolev space which includes unbounded functions. In such cases, one has to use an extension of pointwise smoothness, which was introduced by Calderón and Zygmund in 1961, see [START_REF] Caldéron | Local properties of solutions of elliptic partial differential equations[END_REF]. Let B(x 0 ,R) denote the open ball centered at x 0 and of radius R. We first define the global setting fitted to this notion.

Definition 2 Let f :→ C be a function which locally belongs to L p (IR d ). The function f has slow growth in

L p if ∃A, C > 0, ∀R, B(0,R) |f (x)| p dx ≤ C(1 + R) A . ( 4
)
Definition 3 Let p ∈ [1, +∞) and α>-d/p. Let f be a function with slow growth in L p . We say that f belongs to T p α (x 0 ) if there exist C>0 and a polynomial P of degree less than α such that, for r small enough,

   1 r d B(x 0 ,r) |f (x) -P (x -x 0 )| p dx    1/p ≤ Cr α . ( 5
)
The p-exponent of f at

x 0 is h p f (x 0 ) = sup{α : f ∈ T p α (x 0 )}.
Note that h ∞ f (x 0 )=h f (x 0 ) and, if p ≥ q, then h p f (x 0 ) ≤ h q f (x 0 ), see [START_REF] Caldéron | Local properties of solutions of elliptic partial differential equations[END_REF].

A pointwise irregularity criterium

We start by establishing a general pointwise irregularity criterium based on the Gabor-wavelet transform (referred to in the following as the GW transform). We will use the following notations: If λ, x ∈ IR d , λ•x denotes the usual scalar product of λ and x; if Ω is a rotation in IR d (i.e. belongs to SO d ), then

φ Ω (x)=φ(Ω(x)).
Definition 4 Let φ :I R d -→ IR be a function in the Schwartz class such that φ(ξ) is supported in the unit ball centered at 0. The GW transform of a function or a tempered distribution f defined on IR d is defined by

d(a, b, λ, Ω) = 1 a d IR d f (x)e -iλ•x φ Ω x -b a dx. (6)
The following pointwise irregularity criterium is a variant of the usual wavelet criterium, either for Hölder regularity, see [START_REF] Jaffard | Pointwise smoothness, two-microlocalization and wavelet coefficients[END_REF], or for T p α regularity, see [START_REF] Jaffard | Wavelet Techniques for pointwise regularity[END_REF][START_REF] Jaffard | Wavelet analysis of fractal Boundaries, Part 1: Local regularity and Part 2: Multifractal formalism[END_REF].

Remark: If we pick λ of the form λ = u/a, where u is a fixed vector satisfying u ≥ 1, then [START_REF] Hardy | Some problems of Diophantine approximation[END_REF] boils down to the usual the two-microlocal condition f ∈ C α,-α-d/p (x 0 ); indeed, ψ(x)=e -iux φ(x) has vanishing moments of all orders so that restricting to constants λ of this form in the GW transform amounts to consider the usual continuous wavelet transform of f . This twomicrolocal condition essentially means that a fractional primitive of f of order d/p belongs to C α+d/p (x 0 ), see [START_REF] Jaffard | Pointwise smoothness, two-microlocalization and wavelet coefficients[END_REF][START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF] for precise statements and results, and [START_REF] Jaffard | Wavelet Techniques for pointwise regularity[END_REF] for sharper results in the case of the wavelet transform.

Proof of Proposition 5:

If ω a,λ,Ω (x)=a -d e -iλ•x φ Ω (x/a), then ω a,λ,Ω (ξ)= φ(a Ω * (ξ + λ)),
so that, as soon as |λ| > 1/a, then ω a,λ,Ω and all its derivatives vanishe at 0. It follows that, if P is the polynomial given by (5), then

d(a, b, λ, Ω) = 1 a d IR d (f (x) -P (x -x 0 ))e iλ•x φ Ω x -b a dx.
For n ≥ 0, let B n = B(b, 2 n a), ∆ n = B n+1 -B n and ∆ 0 = B 0 . We split d(a, b, λ, Ω) as a sum of integrals I n over ∆ n . Let q denote the conjugate exponent of p; by Hölder's inequality,

∀n ≥ 0, |I n |≤ 1 a d f (x) -P (x -x 0 ) L p (B n+1 ) φ Ω x -b a L q (∆n)
.

Note that, because of the global slow growth assumption, the precise value of r in ( 5) is irrelevant, and we can pick r = 2.

Let N be such that

2 N a ≤ 1 < 2 N +1 a.
If n ≤ N , then all integrals bear on domains included in B(x 0 , 2). In that case, since B n+1 ⊂ B(x 0 , |x 0 -b| +2 n+1 a), and since φ has fast decay, for all D large enough,

|I n |≤ CC (D) a d (|x 0 -b| +2 n+1 a) α+d/p a d/q (2 -Dn ) 1/q .
If n>N, we use the slow growth assumption on f , which yields that

|I n |≤C(2 n a) A a d/q (2 -Dn ) 1/q ,
where A is given, but D can be picked aribitrarily large (because of the fast decay of φ).

Summing up on all values of n, we obtain that |d(a, b, λ, Ω)| is bounded by

Ca -d/p N n=0 (|x 0 -b| +2 n+1 a) α+d/p 2 -Dnq + ∞ N +1 C(2 n a) A a d/q (2 -Dn ) 1/q ≤ C |x 0 -b| α+d/p a -d/p + a α .
Hence Proposition 5 holds.

A first application to nonharmonic Fourier series

Let (λ n ) n∈IN be a sequence of points in IR d . We will consider series of the form given by [START_REF] Jaffard | Multifractal formalism and anisotropic selfsimilar functions[END_REF]. Of course, we can (and will) assume that the λ n are distinct; note that we do not assume the λ n to be integers. One refers to such series as nonharmonic Fourier series, see [START_REF] Young | A introduction to nonharmonic Fourier series[END_REF]. In order to perform a GW analysis of f , we need this series to converge in the space of tempered distributions. A straightforward sufficient condition for convergence in S of (1) is

∃N ∈ IN : n∈IN |a n | (1 + |λ n |) N < ∞. ( 8
)
Definition 6 Let (λ n ) be a sequence in IR d . The gap sequence associated with (λ n ) is the sequence (θ n ) defined by

θ n = inf m =n |λ n -λ m |
(θ n is the distance between λ n and its closest neighbour).

We say that the sequence

(λ n ) is separated if inf n θ n > 0.
Note that (8) is not a necessary condition for the convergence of (1) in S ; however, if the sequence (λ n ) is a finite union of separated sequences (which will always be the case in the applications that we will consider), then one easily checks that ( 8) is indeed a necessary and sufficient condition of convergence in S . Therefore, we will always assume in the following that (8) holds.

The following statement includes, and slightly improves the previous results concerning everywhere irregularity of (nonharmonic) Fourier series.

Theorem 7 Let f be given by [START_REF] Jaffard | Multifractal formalism and anisotropic selfsimilar functions[END_REF], where we assume that the sequences (λ n ) n∈IN and (a n ) n∈IN satisfy [START_REF] Izumi | Théorèmes élémentaires sur les séries de Fourier lacunaires[END_REF]. Let x 0 be a given point of IR d , p>1, α>-d/p and assume that f has slow growth in L p . If f ∈ T p α (x 0 ), then there exists C such that, for all n,

if |λ n |≥θ n , then |a n |≤ C (θ n ) α . ( 9
)
Thus, if

H = sup{α : (9) holds}, then, for any x 0 ∈ IR d , h p f (x 0 ) ≤ H (hence h f (x 0 ) ≤ H).
Note that previous results anticipating Theorem 7 only dealt with pointwise Hölder regularity. The following definition supplies a strong lacunarity condition, often considered in the past.

Definition 8 Let (λ n ) n∈IN ∈ IR d ; it is a Hadamard sequence if it is separated and satisfies ∃C>0 such that ∀n, θ n ≥ C|λ n |. (10)
Many papers have dealt with irregularity properties of lacunary Fourier series. We only mention a few landmarks that anticipated Theorem 7.

• In 1962, G. Freud considered one-dimensional Hadamard sequences of integer in dimension 1, see [START_REF] Freud | On Fourier series with Hadamard gaps[END_REF]. He showed that, if f ∈ L 1 (T ), then h f is constant. • In 1965, M. Izumi, S.-I. Izumi and J.-P. Kahane obtained Theorem 7 in the following case: d = 1, λ n ∈ ZZ , f is continuous and f ∈ C α (x 0 ), see [START_REF] Izumi | Théorèmes élémentaires sur les séries de Fourier lacunaires[END_REF].

• The first multidimensional result was obtained by J. Pesek in 1977, who considered periodic multiple Fourier series, i.e. when λ n ∈ ZZ d , see [START_REF] Pesek | One point regularity properties of multiple Fourier series with gaps[END_REF] (and also [START_REF] Luther | The differentiability of Fourier gap series and "Riemann's example" of a continuous nondifferentiable function[END_REF] for related results). Assume that f ∈ L 1 (T d ), f ∈ C α (x 0 ) and let ω ∈ (0, 1); J. Pesek showed that, if

∀n, θ n ≥ C|λ n | ω , (11)
then

a n = O |λ n | -ωα
; note that the assumptions of Theorem 7 actually do not require the lacunarity condition to hold uniformly for all the λ n : in order to recover an irregularity result, one only needs the lacunarity condition to hold on a subsequence.

• In 2006, J. Dixmier, J.-P. Kahane and J.-L. Nicolas proved Theorem 7 when the λ n form a Hadamard sequence, see [START_REF] Dixmier | Un exemple de nondérivabilité en géométrie du triangle[END_REF].

Proof of Theorem 7: It is a direct consequence of Proposition 5: We estimate the GW transform of f at particular points, and for a function φ such that φ(ξ) is radial, supported in the unit ball centered at 0, and such that φ(0) = 1.

Taking Ω = Id, we consider

D m = d 1 θ m ,x 0 ,λ m ,Id . (12)
On one hand,

D m =(θ m ) d n a n e i(λn-λm)•x φ(θ m (x -x 0 )) dx = n a n φ λ m -λ n θ n e i(λn-λm)•x 0 ; ( 13
)
since φ vanishes outside of B(0, 1), the definition of θ n implies that φ ((λ mλ n )/θ n )=δ n,m , so that D m = a m . On the other hand, if f ∈ T p α (x 0 ), then Proposition 5 implies that, for any m such that |λ m |≥θ m , |D m |≤Cθ -α m ; Theorem 7 follows.

2 The Optimality of Theorem 7 J. Pesek showed that, for a given sequence of frequencies λ n ∈ ZZ d , and for given α ∈ (0, 1], and ω ∈ (0, 1], if [START_REF] Jaffard | Wavelet Techniques for pointwise regularity[END_REF] does not hold, then there exists a sequence (a n ) such that

a n = O |λ n | -ωα , though f ∈ L 1 (T d ) and f ∈ C α (x 0 ).
We now extend Pesek's optimality theorem in the following direction: the condition λ n ∈ ZZ d is not required (which means that its proof does not require the use of explicit trigonometric polynomials).

Proposition 9 Let λ n be a sequence taking values in IR d . Let α ∈ (0, 1) and ω ∈ (0, 1]. Assume that there exists a subsequence λ n(m) such that [START_REF] Jaffard | Wavelet Techniques for pointwise regularity[END_REF] does not hold; then there exists a sequence

(a n ) such that a n = O |λ n | -ωα , but f ∈ L 1 loc and f ∈ C α (x 0 ).
Proof of Proposition 9: It is clearly sufficient to construct a one-dimensional example, and one can restrict to the case x 0 = 0. After perhaps extracting a subsequence, we can assume that λ n has a subsequence composed of two sets of frequencies µ n and ν n which both increase at least exponentially, and satisfy

|µ n -ν n |≤ε n |µ n | ω
where (ε n ) tends to 0. Let γ n be a sequence satisfying

• γ n is decreasing and tends to 0

• γ n ≥ ε n • γ n+1 /γ n → 1. Let f (x)= n 1 γ α n |µ n | ωα e iµnx -e iνnx . (14)
Note that

a n = O |λ n | -ωα
. Proposition 9 will be proved if we can show that f ∈ C α (x 0 ). The assumptions on γ n and the exponential growth of µ n imply that the sequence 1/(γ n |µ n | ω ) is decreasing for n large enough and tends to 0. Let N be such that

1 γ N +1 |µ N +1 | ω < |x|≤ 1 γ N |µ N | ω . Then |f (x)|≤ N n=1 1 γ α n |µ n | ωα |µ n -ν n ||x| +2 ∞ n=N +1 1 γ α n |µ n | ωα ≤ N n=1 γ 1-α n |µ n | ω(1-α) |x| +2 ∞ n=N +1 1 γ α n |µ n | ωα .
Once again, the assumptions on γ n and the exponential growth of µ n imply that the sequence γ 1-α

n |µ n | ω(1-α) (respectively γ -α n |µ n | -ωα ) increases (respec- tively decreases) geometrically; therefore |f (x)|≤Cγ 1-α N |µ N | ω(1-α) |x| + C 1 γ α N |µ N | ωα ,
which, using the definition of N is bounded by C|x| α . J. Pesek expected his optimality result to be true without the limitation α<1 (and he actually claimed that, in dimension 1, his proof extends to larger values of α). Surprisingly, we will now show that, despite this natural expectation, Theorem 7 is not optimal when α is larger than 1.

Definition 10 Let (λ n ) be a sequence in IR d , with gap sequence (θ n ). The second order gap sequence (ω n ) n∈IN associated with (λ n ) is the distance between λ n and its second closest neighbour; it can be formally defined as follows: Let

A n = {m : |λ n -λ m | = θ n }. If A n has only one element l(n), then ω n = inf m/ ∈{n,l(n)} |λ n -λ m |; else ω n = θ n .
Note that θ n ≤ ω n . The following result improves theorem 7 when θ n = o(ω n ) and α>1; therefore it shows that, in such cases, Theorem 7 is not optimal.

Theorem 11 Let f be given by ( 1), where we assume that the sequences (λ n ) n∈IN and (a n ) n∈IN satisfy [START_REF] Izumi | Théorèmes élémentaires sur les séries de Fourier lacunaires[END_REF]. Let x 0 be a given point of IR d , p>1, and assume that f belongs to L p in a neighbourhood of x 0 . Let α ≥ 1; if f ∈ T p α (x 0 ), then there exists C such that, for all n,

if |λ n |≥ω n , then |a n |≤ C (ω n ) α-1 θ n . ( 15
)
Proof of Theorem 11: First, note that, if θ n is not a o(ω n ), then Theorem 11 boils down to Theorem 7; so, we can assume that

θ n = o(ω n ).
We apply again Proposition 5, but instead of the previous function φ, we use φ 1 defined by φ1 (ξ)= φ(ξ)ξ 1 , where ξ 1 denotes the first component of ξ and φ(ξ) is radial, supported in the unit ball centered at 0, and such that φ(ξ) = 1 in a neighbourhood of 0. Denote by Ω m a rotation which maps the first canonical vector e 1 to the unit vector in the direction of λ mλ l(m) , and let

E m = d 1 ω m ,x 0 ,λ m , Ω m .
A similar computation to the one which led to (13) now yields

D m = a m |λ l(m) -λ m |/ω m ; Proposition 5 implies that |D m |≤Cω -α m ; since |λ l(m) -λ m | = θ m , Theorem 7 follows.
One can easily prove the optimality of Theorem 11 when 1 ≤ α ≤ 2, using a function f similar to [START_REF] Luther | The differentiability of Fourier gap series and "Riemann's example" of a continuous nondifferentiable function[END_REF], but which involves differences of order two of complex exponentials. Note also that one could push the argument further and obtain more complicated criteria that would be optimal for larger and larger values of α.

An unexplored open problem is to understand the optimality of these criteria when both the sequence λ n and the order of magnitude of the a n are given. Let us give a simple example: Let (a n ) be a sequence of coefficients satisfying

∃C, C > 0 such that ∀n ≥ 1, C n 2 ≤|a n |≤ C n 2 , (16)
and let

f (t)= ∞ n=1 a n sin(n 2 t). (17)
Proposition 9 or Theorem 11 yield that the Hölder exponent of f is everywhere smaller than 2. In the case where a n =1/n 2 , the function f considered is (2), and its largest Hölder exponent is 3/2, see [START_REF] Jaffard | The spectrum of singularities of Riemann's function[END_REF]; however, it is not known if this is best possible; i.e. does there exist a sequence (a n ) satisfying [START_REF] Oskolkov | The Schrödinger density and the Talbot effect, Approximation and Probability[END_REF] and such that the Hölder exponent of f at some points is larger than 3/2? Can it be as large as 2? (This is expected if Theorem 11 is optimal in that case.)

3
Applications and extensions of Theorems 7 and 11

3.1 Hölder range of (q, δ) lacunary nonharmonic Fourier series

We first apply Theorems 7 and 11 to a class of lacunary nonharmonic Fourier series whose frequencies grow polynomially, as in (2).

Definition 12 Let f be given by [START_REF] Jaffard | Multifractal formalism and anisotropic selfsimilar functions[END_REF], and let δ be defined by

δ = lim inf -log |a n | log |λ n | .
Assume that δ is finite and that, for some q>1, ∀n,

θ n ≥ C|λ n | (q-1)/q , ( 18
)
then f is called a (q, δ)-series.

The series (2) is an example of (2, 1)-series. Typical examples of (q, δ)-series are supplied by

R(n)e i(Q(n)x) ,
where R and Q are rational fractions, and

q = deg(Q), δ = -deg(R)/deg(Q) (R
and Q can also be rational functions of n, log n, log log n, . . . in which case, only the degrees in the variable n play a role).

Definition 13 Let f ∈ L ∞ loc (IR d ). The Hölder range of f is H R (f )={H : ∃x 0 ∈ IR d : h f (x 0 )=H}.
A function f is a monohölder function if its Hölder range is reduced to one point, i.e. if its Hölder exponent is constant.

Results on the Hölder range will make use of the following uniform regularity criteria.

Lemma 14 Let f be an L 1 loc function given by [START_REF] Jaffard | Multifractal formalism and anisotropic selfsimilar functions[END_REF] with

λ n ∈ IR d . Let θ = sup{ω : a n |λ n | ω ∈ l 1 }.
Then, for all ω<θ, the function f belongs to C ω (IR d ).

If there exist a, b > 0 such that

N n=1 |a n λ n | = O(N a ) and ∞ n=N |a n | = O 1 N b , ( 19
)
then f belongs to C b/(a+b) (IR d ).
Proof of Lemma 14: Assume that θ ≤ 1. Let ω<θ; then |e iλn.

xe iλn.y | is obviously bounded by 2, and (using the mean value theorem) by

|λ n ||x -y|, hence by 2|x -y| ω |λ n | ω ; therefore |f (x) -f (y)|≤2|x -y| ω |a n ||λ n | ω ≤ C|x -y| ω .
The case θ>1 follows by taking partial derivatives of the series (1) up to the order [θ]. The proof of the second part is similar: Let x, y ∈ IR d and N be defined by

1 N a+b ≤|x -y| < 1 (N -1) a+b .
In the estimate of |f (x)-f (y)|, one bounds the increment |e iλn.x -e iλn.y | either by |λ n ||x -y| if n ≤ N or by 2 else.

Note that the second part of Lemma 14 also extends to regularity exponents larger than 1 by derivating the series.

The following corollary gives a quantitative content to the intuitive rule that the more lacunary the Fourier series, the smaller its Hölder range.

Corollary 15 Let f be given by [START_REF] Jaffard | Multifractal formalism and anisotropic selfsimilar functions[END_REF].

• If there exists a subsequence n k such that |a n k |≥C|λ n k | -δ and θ n k ≥ C|λ n k | (q-1)/q , then ∀x ∈ IR d ,h f (x) ≤ δq/(q -1).
• If f is a (q, δ)-series with δ > d/q, its Hölder range is contained in the interval

δ -d q , δq q-1
.

Remark: When q =+∞ (i.e. if the sequence (λ n ) has fast growth), this result boils down to a classical monohölder property of such Fourier series (hence also valid for Hadamard series):

Let δ = lim inf -log(a n ) log(λ n ) ; if δ>0, then ∀xh f (x)=δ.

Proof of Corollary 15:

The first statement follows from Theorem 7. Denote by Λ j the shell

Λ j = {ξ :2 j ≤|ξ| < 2.2 j }.
If (λ n ) satisfies [START_REF] Young | A introduction to nonharmonic Fourier series[END_REF], then the number of frequencies λ n in Λ j is bounded by

N j = C. 2 dj (2 j ) d(q-1)/q = C.2 dj/q .
In order to use Lemma 14, we split the series |a n |||λ n | ω into sums over each shell. The sum for λ n ∈ Λ j is bounded by 2 dj/q 2 -δj 2 -ωj , and therefore, the whole series converges if ω<δ-d/q. Hence the lower bound for the Hölder exponent holds.

A last implication concerns chirp exponents of lacunary Fourier series. We start by recalling the definition of a chirp, see [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF]. Let f be a tempered distribution; fractional primitives of f can be defined as follows: Let F denote the Fourier transform; I s is the operator satisfying

F(I s (f ))(ξ) = (1 + |ξ| 2 ) -s/2 F(f )(ξ).
Definition 16 Let f be a locally bounded function with slow growth;

f is a chirp of type (h, γ) at x 0 if ∀s ≥ 0,I s (f ) ∈ C h+s(γ+1) (x 0 ). ( 20
)
The chirp exponent of f at x 0 is

β f (x 0 ) = sup{γ : (20) holds for h =0}.
Remark: It is sufficient to check that (20) holds for s = 0 and for a sequence s n which tends to +∞, see [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF].

Typical examples of chirps (in dimension d = 1) are supplied by the functions

f α,β (x)=|x| α sin 1 |x| β ,
whose Hölder exponent at 0 is α and chirp exponent is β, see [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF].

Note that (Id-∆ 2 )e iλ•x = (1+|λ| 2 )e iλ•x . It follows that, if f is a (q, δ) lacunary Fourier series, then, it can be written as a fractional derivative of order 2 of a (q, δ+2) lacunary Fourier series, and therefore (using the remark which follows Definition 16) the Hölder exponent of its fractional primitive of order 2 will be at most (δ + 1)q/(q -1). Using this argument for primitives of arbitary order, one obtains the following conclusion.

Corollary 17

The chirp exponent of a (q, δ) lacunary Fourier series is at most 1/(q -1).

Two historical examples show that this result is sharp: (2) has a chirp of exponent 1, see [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF], and D. Boichu proved that n -3 sin(n 3 t) has chirps of exponent 1/2, see [START_REF] Boichu | Analyse 2-microlocal et développement en série de chirps d'une fonction de Riemann et de ses généralisations[END_REF].

Additional irregularity criteria

We will now show the flexibility of the method used in the proof of Theorem 7 by obtaining additional irregularity criteria which depend either on noncancellation properties of the coefficients a n or on particular spacings of the frequencies. We start by defining the gap sequences of arbitrary order associated with a sequence λ n .

Definition 18 Let λ n ∈ IR d , and let k be a positive integer. The gap sequence of order k asociated with λ n is the sequence θ k n such that θ k n is the distance between λ n and its k-th closest neighbour (counted with multiplicity).

The set A k n is composed of the k-closest neighbours of λ n (including λ n ), and

B k n = A k n -{λ n }.

Remarks:

The sets A k n and B k n are not well defined if several frequencies are at the same distance with λ n ; however, in that case, we can make an arbitary choice among these frequencies; it will not have any incidence on the following results. Note that θ 1 n = θ n and θ 2 n = ω n .

Proposition 19 Let f be given by [START_REF] Jaffard | Multifractal formalism and anisotropic selfsimilar functions[END_REF], where the sequences (λ n ) n∈IN and (a n ) n∈IN satisfy [START_REF] Izumi | Théorèmes élémentaires sur les séries de Fourier lacunaires[END_REF]. Let ω<1 be given, and assume that n and k are such that

θ k-1 n ≤ ωθ k n . Let x 0 be a given point of IR d , p>1 and R>0; if f ∈ T p α (x 0 ), then ∃C>0, ∀b ∈ B x 0 , R θ k n , l∈A k n a l e iλ l •b ≤ C(1 + R) α+d/p (θ k n ) α . ( 21
)
If furthermore there exists A>0 such that

|a n |≥(1 + A) l∈B k n a l e iλ l •b , then |a n |≤ C (θ k n ) α .
Proof of Proposition 19: The proof is similar to the proof of Theorem 7, but requires a function φ which satisfies the stronger requirement supp( φ) ⊂ B(0, 1) and

∀ξ such that |ξ|≤ω, | φ(ξ)| =1. (22) Let D n = d 1 θ k n , b, λ n ,Id
. On one hand, since φ satisfies (22), the same argument that led to (13) now yields

D n = l a l φ λ l -λ n θ k n e i(λ l -λn)•b = l∈An k a l e i(λ l -λn)b . ( 23
)
On the other hand, if f ∈ T p α (x 0 ), then Proposition 5 implies that |D n |≤ C(1 + R) α+d/p (θ k n ) -α ; the second part of Proposition 19 follows.

The third part follows immediately, since

l∈A k n a l e iλ l •b ≥|a n |- l∈B k n a l e iλ l •b ≥ A A +1 |a n |.
The second part of Proposition 19 yields the same conclusion as Theorem 7 in settings where no separation condition between the frequencies holds. Unlike Theorems 7 and 11, the first part can give a different upper bound of the Hölder exponent at different points.

In order to illustrate the different ranges of applications of Theorem 7, Theorem 11 and Proposition 19 we compare them on the toy-example supplied by the series

f α (x)= 1 2 ∞ j=0 2 -αj cos((2 j -1)x) -cos((2 j +1)x) = sin x ∞ j=0 2 -αj sin(2 j x),
where α ∈ (0, 1). Indeed, in that case, the local exponents can be computed by hand and one immediately checks that h fα (kπ)=α + 1, and everywhere else: h fα (x)=α. As a test of the respective efficiencies of the previous results, let us check what they yield in this situation.

The frequencies of this series are a union of two Hadamard series. Since forall n, one has θ n = 2, Theorem 7 does not yield any irregularity result. Since ω n =2 j-1 -2, Theorem 11 allows to capture the largest regularity index: It states that f α is nowhere C α+1 . Finally, Proposition 19 allows to discriminate between behaviors at different points: Taking R = 0 in (21) yields that, if f α ∈ C β (x 0 ), then 2 -αj |e i(2 j -1)x 0e i(2 j +1)x 0 |≤C2 -βj ; which implies the sharp upper bound for the Hölder exponent at every point.

Everywhere irregularity of solutions of PDEs

As a consequence of the previous results, we will show that the solutions of some PDEs display a remarkable property of everywhere irregularity if the initial condition is not smooth. Our purpose is not to obtain results in their most general form but, through some toy-examples, to point out unexpected applications of Theorem 7 in the field of PDEs.

Schrödinger's equation in one dimension

Consider the one-dimensional Schrödinger equation without potential

i ∂ψ ∂t = - ∂ 2 ψ ∂x 2 , for (x, t) ∈ IR × IR (24)
with initial condition:

ψ(x, 0) = ψ 0 (x)= n∈ZZ a n e inx . ( 25
)
We first assume that ψ 0 is a periodic distribution, i.e. that the sequence a n increases at most polynomially. Let us make precise the way by which its initial value can be defined. If u(x) and φ(t) are two functions, (u ⊗ φ)(x, t) denotes the function u(x)φ(t).

Definition 20 Let t 0 ∈ IR ; let φ ∈D (IR) be such that

φ(x)dx =1 .L e t φ λ (t)= 1 λ φ t -t 0 λ
; the trace of a two-variables distribution ω at time t 0 (if it exists) is the distribution ω t 0 such that ∀u ∈D(IR)

ω t 0 |u = lim λ→0 ω|u ⊗ φ λ .
The general solution of (24) can be written

ψ(x, t)= n∈ZZ a n e inx e -in 2 t , (26)
which means that (26) has the following properties:

• It is indeed a solution of (24) in the sense of distributions.

• It has a trace ψ t 0 at any time t 0 ∈ IR, in the sense of Definition 20, and at time t 0 = 0, its value is given by (25). Corollary 21 Let s>-5/2, let ψ(x, t) be a solution of ( 24), and assume that ψ 0 / ∈ H s . Then ∀α>s+1/2, ∀(x 0 ,t 0 ), ∀p>1,ψ / ∈ T p α (x 0 ,t 0 ).

Furthermore, as regards irregularity in the time direction, ∀α>s+1/2, ∀(x 0 ,t 0 ), ∀p>1,ψ x 0 / ∈ T p α (t 0 ).

In particular, when (24) has a bounded solution, then ∀α>s+1/2, ∀(x 0 ,t 0 ),

ψ x 0 / ∈ C α (t 0 ).
Remarks: This is an irregularity result either in all directions or in the time direction. The solutions can of course be smooth for a given t 0 in the x direction. For example, assume that the initial condition is piecewise smooth; the solution at time t =2 π reproduces the initial condition, and is therefore piecewise smooth in the x direction.

The everywhere irregularity of solutions of the Schrödinger equation somehow means that their graph is a "fractal"; such properties for the graph of the fundamental solution of (24) have been investigated by K. Oskolkov, see [START_REF] Oskolkov | The Schrödinger density and the Talbot effect, Approximation and Probability[END_REF] and references therein.

Simply supported beams

A closely related equation is supplied by the vibrations of simply supported beams:

if (x, t) ∈ [0,π] × IR + , ∂ 2 u ∂t 2 + ∂ 4 u ∂x 4 =0, ( 29
) if x ∈ [0,π],u (x, 0) = u 0 (x) and ∂u ∂t (x, 0) = u 1 (x), ( 30
) if x =0, and x = π, u(0,t)=u(π, t)=0. ( 31
)
The general solution can formally be written

u(x, t)= ∞ n=1 (a n cos(n 2 t)+b n sin(n 2 t)) sin(nx),
where the a n and b n are defined by: u 0 (x)= a n sin(nx), and u 1 (x)= n 2 b n sin(nx). The solution is of the same type as (26), except that we have to add the frequencies µ n =( n, n 2 ); however, the associated gap sequence θ n is the same. Denote by u x 0 the solution at time t 0 ; since the beam equation makes sense only if the solution is continuous, we assume that u 0 ∈ H 1/2 and u 1 ∈ H -3/2 . The same arguments developped for the Schrödinger equation yield the following result.

Corollary 22 Let s>1/2, let u(x, t) be a solution of (29), and assume that either u 0 / ∈ H s or u 1 / ∈ H s-2 . Then ∀α>s+1/2, ∀(x 0 ,t 0 ), ∀p>1,u / ∈ T p α (x 0 ,t 0 ), and u x 0 / ∈ T p α (t 0 ).

Schrödinger's equation in two dimensions and vibrating plates

In the previous cases, all frequencies satisfy the uniform separation condition (27); therefore, if the solution is irregular at one point, then all coefficients a n have to be large. We will now consider cases where the separation condition only holds for subsequences of frequencies. In that case, smoothness at one point only implies that the the corresponding subsequence of the a n is small, and one cannot infer everywhere irregularity results for all possible irregular initial condition, as in Corollaries 21 and 22. However, some initial conditions must be smooth, and therefore, by contraposition some initial conditions generate everywhere irregular solutions. Though we won't investigate this topic which leads to intricate problems related with number theory, we just work out two simple two-dimensional examples.

Consider the two-dimensional periodic Schrödinger equation

                 i ∂ψ ∂t = -∆ψ, ψ(x, y, 0) = ψ 0 (x, y)= (n,m)∈ZZ 2 a n,m e i( nx A + my B ) . (32)
We consider distribution solutions in the same sense as in Section 4.1. The general solution is

ψ(x, y, t)= a n,m e i( nx A + my B ) e -i( n 2 A 2 + m 2 B 2 )t = a n,m e iλn,m•X , where λ n,m =( n A , m B , -n 2 A 2 -m 2 B 2 )
, and X =( x, y, t). After an affine transformation, this sequence of frequencies is of the form (n, m, an 2 + bm 2 ), with a, b > 0. The following lemma gives a lower bound for the corresponding gap sequence.

Lemma 23 Let λ n,m =( n, m, an 2 + bm 2 ), with a, b > 0, and (n, m) ∈ ZZ 2 . For m = n 2 , ∃C, ∀n, θ n,n 2 ≥ C λ n,n 2 1/4 (33)
Proof of Lemma 23: For couples (n, m) satisfying m = n 2 , let us estimate ∆=(an 2 + bm 2 )a(n + p) 2b(m + q) 2 , where (p, q) = (0, 0).

In order to prove (33), we will check that there exists C>0 such that either |p|, |q| or |∆| is larger than C|n|; indeed, this will imply that if (p, q) = (0, 0), then λ n,n 2λ p,q ≥ C|n|; since λ n,n 2 is of the order of magnitude of n 4 , (33) will follow.

Note that

∆=-2anp -2bn 2 q -ap 2 -bq 2 ,
where p and q are integers and do not vanish simultaneously.

First assume that q = 0. Then, since p = 0,

|∆| = |ap(2n + p)|≥|a(2n + p)|
so that, either |p|≥|n| or |∆|≥|an|, and (33) holds since λ n,n 4 ∼ n 4 . Now, assume that q = 0. It is sufficient to prove that

|∆| + ap 2 + bq 2 ≥ cn 2 . (34)
If |p|≥| bn 2a |, the result holds. Else, the left hand side of (34) is larger than |b|n 2 /2.

It follows from Theorem 7 that, if f ∈ T p α (x 0 ,t 0 ), then a n,n 2 = O(n| -α ).
The following corollary thus follows.

Corollary 24 let ψ be a solution of (32), where the sequence a n,n 2 is not a O(n| -α ). Then ∀(x 0 ,t 0 ), ∀p>1, ∀α>-d/p, ψ / ∈ T p α (x 0 ,t 0 ). Now, we consider the two-dimensional supported plate equation, in a rectan-

gular domain Ω = [0,A] × [0,B].                ∂ 2 u ∂t 2 +∆ 2 u =0, u =∆u =0 on ∂Ω, u(x, y, 0) = u 0 (x, y) and ∂u ∂t (x, y, 0) = u 1 (x, y). (35)
The general solution can formally be written

u(x, y, t)= sin nx A sin my B a n,m cos n 2 A 2 + m 2 B 2 t + b n,m sin n 2 A 2 + m 2 B 2 t = c l e iλ l •X ,
where the λ l take all values of the form

( n A , m B , ±( n 2 A 2 + m 2 B 2 )
), and X =(x, y, t). This is almost the same set of frequencies as in the Schrödinger case, and one immediately checks that it satisfies the same gap condition. The following result follows.

Corollary 25 Let u(x, y, t) be a solution of (35) with initial conditions

u(x, y, 0) = a n,m sin nx A sin my B , ∂u ∂t (x, y, 0) = b n,m n 2 A 2 + m 2 B 2 sin nx A sin my B . If the sequence a n,n 2 is not a o(|n| -α ) or if the sequence b n,n 2 is not a o(|n| -α ), then ∀(x 0 ,t 0 ), ∀p>1, ∀α>- d p ,ψ / ∈ T p α (x 0 ,t 0 ).
Remark: Clearly, many other subsequences than (n, n 2 ) would lead to the same conclusion. We picked it as an illustration, but it may be changed depending on the particular initial conditions considered.

Directional irregularity of lacunary Fourier series

We will now extend Theorem 7 in a non-isotropic setting in order to obtain directional irregularity results for multidimensional lacunary Fourier series. We start by defining and motivating the notion of directional regularity that will be used.

Directional regularity

Let f :I R d -→ C be a locally bounded function. The definition of pointwise regularity supplied by Definition 1 is uniform in all directions; in order to take into account directional behaviors, it is natural to define the Hölder regularity at x 0 in a direction u ∈ IR d -{0} as the Hölder regularity at 0 of the one variable function t → f (x 0 + tu). This definition has several drawbacks which are the consequence of the fact that, actually, it is defined as the trace of f on a line, which is a set of vanishing measure. Indeed, one ultimately wants to deduce this directional smoothness from the coefficients of f on some particular sets of functions (ridgelets or curvelets for instance). These functions will have a support of non-empty interior, and therefore will take into account the values of f around the line considered. Therefore the definition of directional smoothness should include such information. However, in the asymptotic of small scales, the values taken into account should be localized more and more sharply around this line. These considerations motivate Definition 27 below. We start by extending the definition of the degree of a polynomial in an anisotropic setting.

Definition 26 Let α =(α 1 , •••α d ) be d-
C α • t a 1 1 •••t a d d satisfies a i α i < 1.
In the isotropic case where all the α i are equal, this definition boils down to the usual definition of the degree. Let U =( u 1 , •••,u d ) be an orthonormal basis of IR d ; we denote by (t 1 , •••,t d ) the coordinates of xx 0 on the basis U, i.e.

x -

x 0 = d i=1 t i u i . ( 36
)
We now give the definition of anisotropic smoothness, which was already introduced by M. Ben Slimane in [START_REF] Jaffard | Multifractal formalism and anisotropic selfsimilar functions[END_REF]

|t i | α i . (37)
The degree condition imposed on P implies uniqueness, as a consequence of the following lemma.

Lemma 28 Let α =(α 1 , •••α d ) be d-uple of positive real numbers. If a polynomial P has degree less than α and satisfies

|P (x)|≤C d i=1 |x i | α i , (38)
for x small enough, then P =0.

Proof: We write P (x)= C a x a . We consider P in the cone defined by: ∀i, x i > 0, where we perform the changes of variable t i = x a i i .

Let ω i (a)=a i /α i and

f (t)= a C a i t ω i (a) i
; then (38) becomes |f (t)|≤C|t|, Writing t in polar coordinates, we get

f (t)= ωa r ω(a) g ω(a) (t),
where ω(a)= ω i (a) and the g ω(a) are homogeneous of degree 0. Ordering the ω(a) by increasing order, one obtains by induction that each g ω(a) vanishes, which implies that each term C a x a vanishes (where the sum is taken on the indices a such that ω(a) is given). Since the cone considered has a nonempty interior, this implies that the corresponding polynomial vanishes, and therefore that, forall a, C a = 0.

Let α =( α 1 , •••α d ) is a d-uple of
nonnegative real numbers. We define the average regularity α as the harmonic mean of the α i , i.e.

1 α = 1 d d j=1 1 α j .
The anisotropy indices ν i are the ν i =α/α i . Therefore, ν i = d; note that, in the anisotropic case, ∀i, ν i = 1. The anisotropic pointwise regularity can also be interpreted as a uniform condition inside some elongated ellipsoïds.

Definition 29 The ε-neighbourhood of x 0 of directions U and exponent α, denoted by N ε U , is the set of points x whose coordinates on the basis U satisfy

d i=1 t i ε ν i 2 ≤ 1.
One easily checks that f belongs to C α (x 0 , U) if and only if, for ε small enough, ∃C>0, ∃P : deg(P ) <α and sup

x∈N ε U ,α |f (x) -P (x -x 0 )|≤Cε α. ( 39
)
Remarks:

• The usual pointwise Hölder condition corresponds to the case

α 1 = ••• = α d = α. • If (37) holds, then the one dimensional function t → f (x 0 + tu i ) belongs
to C α i (0) so that, in each direction u i , f has Hölder regularity α i ; thus Definition 27 recaptures the intuitive notion we started with. • In Definition 27, one could start with a nonorthonormal basis U; however, one easily checks that it would not lead to a more general definition of directional smoothness. Indeed, it would be equivalent to the definition we gave, the corresponding orthonormal basis being obtained from the nonorthonormal one through a Gram-Schmidt orthonormalization procedure, starting with the direction of highest regularity and going down through the vectors attached to smaller and smaller regularity indices.

The definition of directional smoothness supplied by (39) has the additional advantage of supplying a starting point for the definition of the directional smoothness of measures and also for an extension of the T p α (x 0 ) regularity criterium to an anisotropic setting.

• A measure µ belongs to M α (x 0 , U) if and only if, for ε small enough,

∃C>0, such that µ(N ε U ,α ) ≤ Cε α. (40)
• A function f belongs to T p α (x 0 , U) if f has slow growth in L p and if there exist C>0 and a polynomial P of degree less than α such that

∀r ≤ R,    1 ε d N ε U |f (x) -P (x -x 0 )| p dx    1/p ≤ Cε α. (41)
Let us now check on a simple example what this notion yields. Let Γ be a smooth curve in IR 2 . We consider the function

f α (x)=(dist(x, Γ)) α .
It is intutively clear that, at a point of Γ, f α has a singularity of order α in the direction orthogonal to Γ, and is smoother in the tangent direction. Indeed, using these directions as basis, one immediately obtains that, at a generic point where the curvature does not vanish,

dist(x, Γ) α ≤ C(t 2α 1 + t α 2 )
(where t 1 and t 2 are respectively the coordinates on the tangent and normal vectors) and this result is optimal. Therefore, at a generic point of Γ, f has Hölder regularity (2α, α). In this case, the ε-neighbouhoods have parabolic scaling, which gives an additional explanation for the special role played by parabolic scalings, as in the FBI transform, the Hart-Smith transform, or curvelets. Note however that, at inflexion points, f has Hölder regularity (3α, α), and it may even display higher anisotropy, depending on the order of tangency between Γ and its tangent.

The notion of Hölder regularity supplied by Definition 27 does not allow to define directly an Hölder exponent as the "best possible" d-uple

(α 1 , •••α d )
such that f belongs to C α (x 0 , U). Indeed this regularity notion does not induce a total order on d-uples, as shown by the example f (x, y)=|xy| whose pointwise (anisotropic) smoothness at 0 is C 2 (0). Clearly, ∀β>0,f ∈ C (1,β) (0,E), and f belongs to C (β,1) (0,E), where E is the canonical basis of IR 2 , and the anisotropic smoothness implies that f belongs to C (2,2) (0,E). However these results clearly do not imply an optimal directional smoothness statement encapsulated in one two-variables exponent. The "compatibility relationships" which hold in general between directional regularity conditions are summarized by a partial ordering property:

If α 1 ≤ β 1 , •••,α d ≤ β d , then f ∈ C β (x 0 , U)= ⇒ f ∈ C α (x 0 , U).
One can define directional regularity exponents in the following way, which is more coherent with our initial motivation.

Definition 30 Let u ∈ IR d -{0}. The Hölder exponent of f in the direction u at x 0 is h f (x 0 ,u) = sup{α : ∃ε>0 f ∈ C (α,ε,•••,ε) (x 0 , U)},
where U is an orthonormal basis starting with the vector u.

Using this definition, the function |xy| we already considered has Hölder exponent +∞ along the two coordinate axes, which is natural to expect.

A criterium of pointwise directional irregularity

We establish a sufficient directional pointwise irregularity condition based on the anisotropic Gabor-Wavelet transform (which we will shorten in aniset transfom). If U is an orthonormal basis of IR We split the integral in two terms. The first one corresponds to the ball |bx 0 |≤1, where we can use (37) (because, since f has slow growth, the constant R in (37) can be chosen arbitraily); it follows that the corresponding integral is bounded by

C |t i | α i φ t 1 -b 1 a ν 1 , ••• t d -b d a ν d dt.
making the change of variable u i =(t ib i )/a ν i , we get |d(a, b, U,ν,λ)|≤C

a ν i α i |u i | α i + |b i | α i |φ(u 1 , •••u d )|dt ≤ CC a α + |b i | α i .
The second integral is bounded using the fast decay of φ, and the slow growth of f , as in the proof of Proposition 5.

In order to obtain an extension of Theorem 7 in the anisotropic setting, we have to define directional gap sequences.

Definition 33 Let (λ n ) be a sequence in IR d , let U be an orthonormal basis of IR d , and let ν be a d-uple satisfying (42). The directional gap θ n of direction U and excentricity ν is

θ n = sup θ : λ n + N θ U {λ m } m =n = ∅ .
Theorem 34 Let f be given by ( 1), where we assume that the sequences (λ n ) n∈IN and (a n ) n∈IN satisfy [START_REF] Izumi | Théorèmes élémentaires sur les séries de Fourier lacunaires[END_REF], and let x 0 ∈ IR d . If f belongs to C α (x 0 , U), then

∃C>0, ∀n ∈ IN if λ n / ∈N θn U then |a n |≤ C (θ n ) α . ( 46
)
Proof of Theorem 34: It is a direct consequence of Theorem 32: We estimate the aniset transform of f at particular points; consider

D m = d 1 θ m ,x 0 , U,ν,λ m . (47)
On one hand,

D m =(θ m ) d n
a n e i(λn-λm)•x φ 1/θm,ν (Ω U (xx 0 ))

dx = n a n φ (λ m ) 1 -(λ n ) 1 (θ m ) ν 1 ••• (λ m ) d -(λ n ) d (θ m ) ν d e iΩ U (λn-λm)•x 0 , (48)
where (λ m ) i is the i-th component of λ m on the basis U. Since φ vanishes outside of B(0, 1), the definition of θ n implies that

φ (λ m ) 1 -(λ n ) 1 (θ m ) ν 1 ••• (λ m ) d -(λ n ) d (θ m ) ν d = δ n,m ,
so that D m = a m . On the other hand, if f ∈ C α (x 0 , U), then Theorem 32 implies that, for any m such that λ n / ∈N θn U , |D m |≤Cθ - α m ; Theorem 34 follows.

a d φ t 1 a ν 1 ,

 1 d , we denote by Ω U the linear mapping that maps the canical basis of IR d to U.L e tν =( ν 1 , •••,ν d )b ea d-uple satisfying ∀i, ν i > 0 andν i = d.(42)We defineφ a,ν (t)= 1 •••, t d a ν d .Definition 31 Let f be a locally bounded function with slow growth. Let φ : IR d -→ IR be a function in the Schwartz class such that φ(0) = 1 and φ is supported in the unit ball centered at 0. The aniset transform of f of direction U and excentricity ν isd(a, b, U,ν,λ)= IR d f (x)e -iλ•x φ a,ν (Ω U (xb)) dx. (43)The anisotropic wavelets φ a,ν (Ω U (xb)) are called "anisets". The following irregularity criterium extends Proposition 5 in the anisotropic setting.Theorem 32 Let f be a locally bounded function with slow growth. If f ∈ C α (x 0 , U), then there exists C>0 such that, ifd i=1 (a ν i |λ i |) 2 ≥ 1, (44)and if a ≤ 1 and |bx 0 |≤1, then then |d(a, b, U,ν,λ)|≤C a α + d i=1 |(x 0 ) ib i | α i .(45)Proof of Theorem 32: After performing on f a translation of x 0 and applying the isometry Ω U , we can assume that x 0 = 0 and that U is the canonical basis of IR d . The Fourier transform of φ a,α is φ (ξ 1 a ν 1, •••,ξ d a ν d ), therefore, it vanishes in a neighbourhood of λ if (44) holds. Thus, in that case, d(a, b, U,ν,λ)= IR d(f (x) -P (x))e -iλ•x φ a,ν (xb) dx.

•

  The trace ψ t 0 is a continuous function of t 0 , i.e. t 0 → ψ t 0 belongs to C(IR, D ); e iλn•X where λ n =(n, -n 2 ), and X =(x, t), so that the gap sequence θ n associated with λ n satisfies But, if |a n |≤C(|n| + 1) -α , then ψ 0 (x) belongs to the periodic Sobolev space H s , as soon as s<α-1/2. (Note that Theorem 11 does not yield any improvement in this case.) One can also consider the trace ψx 0 of ψ at a given point x 0 , as a function of t, i.e. formally, ψx 0 (t)=ψ(x 0 ,t). It is again well defined in the sense of Definition 20, and the solution is still a one-dimensional lacunary Fourier series with λ n = n 2 . we obtain that, if ψx 0 (t) ∈ T 1 α (t 0 ), then |a n |≤C/n α . Hence the following corollary holds.

	Note that (26) is of the form
	∀n ∈ ZZ , θ n ≥|n| +1.				(27)
	Let p>1 and α ≥-2/p. It follows from Theorem 7 that
	if ψ ∈ T p α (x 0 ,t 0 ), then	∀n,	|a n |≤	C (|n| +1) α .	(28)

furthermore, if ψ 0 ∈ H s , for an s ∈ IR, then t 0 → ψ t 0 belongs to C(IR,H s ). n∈ZZ a n

  uple of positive real numbers. A polynomial P has degree less than α if each monomial component of P of the form

  in the case where U is the canonical basis of IR d . Definition 27 Let f :I R d → IR be a locally bounded function with slow growth. Let α =(α 1 , •••α d ) be a d-uple of nonnegative real numbers satisfying α 1 ≥ α 2 ≥•••≥α d .We say that the function f belongs to C α (x 0 , U) if there exists C>0 and a polynomial P of degre lesss than α such that the coordinates t=(t 1 , •••,t d ) ∈ IR d of xx 0 on U satisfy

		d
	∃R>0 such that, if |t|≤R, then	|f (t) -P (t)|≤C
		i=1

Acknowledgement: I thank the anonymous referees for correcting several mistakes and making many suggestions of improvements on the initial version of this paper.


Yvon Maday

Anthony T Patera

G Turinici

Global a Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Symmetric Coercive Elliptic Partial Di erential Equations

We consider \Lagrangian" reduced-basis methods for single-parameter symmetric coercive elliptic partial di erentialequations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced{basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.

R esultats globaux a priori pour l'approximation d' equations aux d eriv ees partielles coercives sym etriques elliptiques d ependant d'un param etre R esum e. On consid ere des m ethodes de bases r eduites de type Lagrange pour des equations aux d eriv ees partielles coercives sym etriques elliptiques et d ependant d'un param etre. On montre que, pour une r epartition logarithmiquement quasi uniforme des points d' echantillonage, l'approximation en base r eduite converge de fa con exponentielle vers la solution exacte uniform ement par rapport au param etre. De plus la convergence ne d epend que faiblement du rapport entre les coe cients de coercivit e et de continuit e de l'op erateur: ainsi une approximation de tr es basse dimension procure une solution tr es pr ecise même dans le cas d'un large eventail de param etres. Des test num eriques (report es par ailleurs) corroborent ces pr edictions num eriques Version fran caise abr eg ee Dans un espace de Hilbert H, muni du produit scalaire ( ; ) Y et de la norme k k Y on se pose le probl eme de trouver u 2 Y v eri ant (1) o u la forme bilin eaire a: Y Y D ! IR d epend d'un param etre 2 D 0; max ]. Sous des conditions classiques de continuit e et de coercivit e de a ce probl eme poss ede une solution unique. La m ethode de base r eduite consiste alors a choisir un entier N et un jeux de param etres S N = f 1 ; : : :; N g pour lesquels, de fa con pr ealable, on calcule | le plus exactement possible | les solutions associ ees u( k ); k = 1; : : :; N.Puis on r esout le syst eme (2) o u W N = Vect fu( k ); k = 1; : : :; Ng: On analyse dans cette note le cas d'un probl eme d ependant d'un seul param etre du type (3) o u a 0 : Y Y ! IR et a 1 : Y Y ! IR sont continues, sym etriques, semi positives et de plus o u a 0 est coercive induisant une norme jjj jjj 2 = a 0 ( ; ) equivalente a celle de Y . Des exemples de probl emes entrant dans ce cadre sont pr esent es, analys es et simul es sur base r eduite dans 12]. Plus particuli erement nous montrons ici que la convergence de cette m ethode en base r eduite est une fonction exponentiellement d ecroissante en le cardinal de W N , et ce uniform ement par rapport au param etre. En particulier on a la borne suivante entre la solution exacte u( ) et son approximation u N ( ) : il existe un entier N crit tel que pour tout N N crit , on a (19) avec une constante c ne d ependant que des conditions d'ellipticit e de a 0 et de max .

La d emonstration de ce r esultat repose d'une part sur le lemme classique de Cea rappel e en (10) et une estimation a priori de la meilleure approximation donn ee dans le lemme 2.

Il convient de noter que l'analyse de la meilleure approximation fait ici intervenir une approximation polynomiale de la solution, mais cette approximation polynomiale est propos ee apr es un changement de variable appropri e ( = e ~ 1 ). Le point qui doit être not e est que la m ethode de Galerkin propose naturellement une approximation dans W N qui est ( a une constante multiplicative pr es) aussi bonne que cette approximation polynomiale en une variable a d e nir. Ceci donne une sup eriorit e et un caract ere g en eral a l'approche variationelle par rapport a une \simple" interpolation puisque aucune connaissance a priori de la forme de la solution en son param etre n'est a conna^ tre.

L'analyse faite ici sugg ere une r epartition logarithmique du jeux de param etres qui donne en e et de meilleurs r esultats dans les applications comme cel a est report e dans 15]. On renvoit aussi a 12] pour plus de d etails sur la mise en oeuvre et les applications.

Introduction

Let Y be an Hilbert space with inner product and norm ( ; ) Y and k k Y = ( ; ) 1=2 Y , respectively.

Consider a parametrized \bilinear" form a: Y Y D ! IR, where D 0; max ], and a bounded linear form f: Y ! IR. We introduce the problem to be solved: Given 2 D, nd u 2 Y such that a(u( )

; v; ) = f(v); 8 v 2 Y :
(1) Under natural conditions on the bilinear form a (e.g. continuity and coercivity) it is readily shown that this problem admits a unique solution.

We introduce an approximation index N, the parameter sample S N = f 1 ; : : :; N g, and the solutions u( k ); k = 1; : : :; N, of problem (1) for this set of parameters. We next de ne the reducedbasis approximation space W N = span fu( k ); k = 1; : : :; Ng: Our reduced-basis approximation is then: Given 2 D, nd u N ( ) 2 W N such that a(u N ( ); v; ) = f(v); 8 v 2 W N :

(2) This discrete problem is well posed under the same former continuity and coercivity conditions.

The reduced-basis approach, as earlier developped, is typically local in parameter space in both practice and theory 1, 2, 4, 9, 10, 13]. To wit, the k are chosen in the vicinity of a particular parameter point and the associated a priori convergence theory relies on asymptotic arguments in su ciently small neighborhoods of 4]. In this note we present, for single-parameter symmetric coercive elliptic partial di erential equations, a rst theoretical a priori convergence result that demonstrates exponential convergence of reduced-basis approximations uniformly over an extended parameter domain. The proof requires, and thus suggests, a point distribution in parameter space which does, indeed, exhibit superior convergence properties in a variety of numerical tests 15]. We refer also to [START_REF] Fink | On the error behaviour of the reduced basis technique for nonlinear nite element approximations[END_REF][START_REF] Machiels | Output bounds for reduced-basis approximations of symmetric positive de nite eigenvalue problems[END_REF][START_REF] Maday | Blackbox reduced-basis output bound methods for shape optimization[END_REF]12] for further discussions of these results and related work and applications.

Problem Formulation

Let us de ne the parametrized \bilinear" form a: Y Y D ! IR as a(w; v; ) a 0 (w; v) + a 1 (w; v) ; [START_REF] Barrett | On the reduced basis method[END_REF] where the bilinear forms a 0 : Y Y ! IR and a 1 : Y Y ! IR are continuous, symmetric and positive semi-de nite; suppose moreover that a 0 is coercive, inducing a (Y -equivalent) norm jjj jjj 2 = a 0 ( ; ). It follows from our assumptions that there exists a real positive constant 1 such that

0 a 1 (v; v) a 0 (v; v) 1 ; 8 v 2 Y : (4)
For these hypotheses, it is readily demonstrated that the problem (1) has a unique solution. Many situations may be modeled by our rather simple problem statement (1), (3). For example, if we take Y = H1 0 ( ) where is a smooth bounded subdomain of IR d=2 , and set a 0 (w; v) = R rw rv, a 1 = R wv, we model conduction in thin plates; here represents the convective heat transfer coe cient. Other choices of a 0 and a 1 can model variable rectilinear geometry, variable orthotropic properties, and variable Robin boundary conditions.

The space Y is typically of in nite dimension so u( ) is, in general, not exactly calculable.

In order to construct our reduced-basis space W N , we must therefore replace u( ) 2 Y by a \truth approximation" u N ( ) 2 Y N Y , solution of the Galerkin approximation a(u N ( ); v; ) = f(v); 8 v 2 Y N : Here Y N , of nite (but typically very high) dimension N, is a su ciently rich approximation subspace such that jjju( ) u N ( )jjj is su ciently small for all in D; for example, for Y = H 1 0 ( ) we know that, for any desired " > 0, we can indeed construct a nite-element approximation space, Y N(") , such that jjju( ) u N(") ( )jjj ".

It shall prove convenient in what follows to introduce a generalized eigenvalue problem: Find (' N i 2 Y N ; N i 2 IR), i = 1; : : :; N, satisfying a 1 ('

N i ; v) = N i a 0 (' N i ; v), 8 v 2 Y N .
We shall order the (perforce real, non-negative) eigenvalues as 0 N N N N 1 N 1 1 , where the last inequality follows directly from (4). We may choose our eigenfunctions such that a 0 (' N i ; ' N j ) = i j ;

(5) and hence a 1 (' N i ; ' N j ) = N i i j , where i j is the Kronecker-delta symbol; and such that Y N can be expressed as span f' i ; i = 1; : : :; Ng. Note that, thanks to the nite dimension of our approximation space Y N , we preclude (the complications associated with) a continuous spectrum | and, as we shall see, at no loss in rigor. We conclude this section by noting that, if we set f N i = f(' N i ), then u N ( ) can be expressed as

u N ( ) = N X i=1 f N i ' N i 1 + N i ; (6)

A Priori Convergence Theory

We propose here to choose the sample points k , k = 1; : : :; N, log-equidistributed in D, where N = ln( max + 1)=N, and is any nite upper bound for 1 1 . Here ~ kN = N c ; 8k; k = 1; : : :; N, and also P N `=1 ~ `N = ln( max + 1) , where c is a real positive constant. Denote the reduced-basis approximation space as W N N = span fu N ( k ); k = 1; : : :; Ng. Although in general dim(W N N ) N, we can suppose that dim(W N N ) = N (otherwise we eliminate elements from W N N until it contains only linearly independent vectors). Then, the (reduced basis)

problem is : Given 2 D, nd u N N ( ) 2 W N N such that a(u N N ( ); v; ) = f(v); 8 v 2 W N N : (7)
This problem admits a unique solution.

Our goal is to (sharply) bound jjju N ( ) u N N ( )jjj, for all 2 D, as a function of N (and ultimately N as well). This error bound in the energy norm can be readily translated into error bounds on continuous-linear-functional outputs 12]; we do not consider this extension further here.

We shall need two standard results from the theory of Galerkin approximation of symmetric coercive problems 14]:

a(u N u N N ; u N u N N ; ) = inf w N N 2W N N a(u N w N N ; u N w N N ; ) ; (8)
a(u N ; u N ; ) a(u; u; ) : [START_REF] Maday | A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Di erential Equations[END_REF] From the positive semide niteness of a 1 , (3), ( 4) and ( 8) we can write

jjju N ( ) u N N ( )jjj 2 a(u N ( ) u N N ( ); u N ( ) u N N ( ); ) inf w N N 2W N N a(u N ( ) w N N ; u N ( ) w N N ; ) (1 + max 1 ) inf w N N 2W N N jjju N ( ) w N
N jjj 2 ; 8 2 D: [START_REF] Noor | Reduced basis technique for nonlinear analysis of structures[END_REF] Also from the de nition of the jjj jjj norm and the positive semide niteness of a 1 , ( 3), ( 4) and ( 9), we obtain jjju N ( )jjj (1 + max 1 ) 1=2 jjju( )jjj; 8 2 D:

(11) We rst state a preparatory result (see 8] for the proof) Lemma 1. Let g(z; ) = 1 1 + e z for z 2 Z ln( 1 ); 1] and 2 0; ] (recall is our strictly positive upper bound for 1 ). Then, for any q 0, jD q 1 g(z; )j 2 q q! ; 8 z 2 Z; 8 2 ; where D q 1 g denotes the q th -derivative of g with respect to the rst argument. We now prove a bound for the best approximation result in Lemma 2. Proof. To facilitate the proof, we shall e ect a change of coordinates in parameter space. To wit, we let e D ln 1 ; ln( max + 1 )], and introduce : e D ! D as (~ ) = e ~ 1 so that 1 ( ) = ln( + 1 ). We then set ũ(~ ) = u( (~ )), ũN (~ ) = u N ( (~ )), and ũN

N (~ ) = u N N ( (~ )). We note that ũN (~ ) = N X i=1 f N i ' N i 1 N i + N i e ~ = N X i=1 f N i ' N i g(~ ; N i ); (13) from (6)
, our change of variable, and the de nition of g. We now observe that in our mapped coordinate, the sample points ~ k 1 ( k ), k = 1; : : :; N, ), n = 1; : : :; M. Note that e I ~ M 1 g( ; )]( 1 ( )) is not a polynomial in . It now follows from ( 5), ( 6), ( 13) and ( 15

We now assume that c N 2 ~ and ~ 1 2 ; under these conditions (recall ( 14)) we obtain (2 ~ ) M ~ ( ~ ; N ) (2 ~ ) ~ =c N , and hence, from ( 16) and ( 17 Then, from (10),(11), Lemma 1, and Lemma 2, we obtain Theorem 3. For N N crit c e ln( max + 1), jjju N ( ) u N N ( )jjj (1 + max 1 ) 1=2 jjju N (0)jjj e N=Ncrit ; 8 2 D; furthermore for N(") such that jjju( ) u N(") ( )jjj ", jjju( ) u N(") N ( )jjj " + (1 + max 1 ) jjju(0)jjj e N=Ncrit ; 8 2 D: Remark 4. By letting " go to zero, we also have jjju( ) u N ( )jjj c jjju(0)jjj e N=Ncrit ; 8 2 D;

(19) for any N N crit with a constant c that depends only on 1 and max .

Remark 5. It must be pointed out that the analysis of the best t in lemma 2 involves a simple polynomial approximation of the solution, but this is a polynomial in the ~ variable. The Galerkin approximation provides this best t, up to a multiplicative constant, regardless of any a priori knowledge of the dependance of the solution on the parameter. This demonstrates the superiority of the reduced basis method with respect to a \simple" interpolation approximation.

  For N N crit c e ln( max + 1

  are equi-distributed with separation ~ k+1 ~ k ' ln( max + 1)=N. It thus follows that, given any ~ 2 D, we can construct a closed interval e I ~ ~ of length ~ that includes ~ and M ~ ( ~ ; N ) distinct points ~ P ~ n , n = 1; : : :; M. Here M ~ ( ~ ; N ) is of the order of ~ N ; more precisely, , we shall often abbreviate M ~ ( ~ ; N ) as M.Now, for any ~ 2 e D, we introduce û~ 2 ( ; ) is the (M 1) th -order polynomial interpolant of g( ; ) through the ~ P ~ n , n = 1; : : :; M; more precisely,

  ( ; )] (~ )j jjju N (0)jjj :(16) We next invoke the standard polynomial interpolation remainder formula 3] and Lemma 1 z; )j ~ M (2 ~ ) M ~ ( ~ ; N ) :

  ), we can write jjjũ N (~ ) û~ jjj jjju N (0)jjj(2 ~ ) ~ =c N : (18) It remains to select a best ~ satisfying c provide the sharpest possible bound, we choose ~ = ~ 1 2e , the minimizer (over all positive ~ ) of (2 ~ ) ~ = N . Our conditions on ~ are readily veri ed: c N 2~ follows directly from the hypothesis of our lemma, N N crit ; and ~ 1 2 follows from inspection. We now insert ~ = ~ into (18) to obtain jjjũ N (~ ) û~ jjj jjju N (0)jjj e N=Ncrit ; for all ~ 2 e D : It immediately follows that, for any 2 D, 1 ( )) û 1 ( ) jjj jjju N (0)jjj e N=Ncrit since û 2 W N N and, for 2 D, 1 ( ) 2 e D. This concludes the proof.

Note that 1 , , and hence S N , are independent of N.

Acknowledgements. We would like to thank Christophe Prud'homme, Dimitrios Rovas, and Karen Veroy of MIT for sharing their numerical results prior to publication. This work was performed while ATP was an Invited Professor at the University of Paris VI in February, 2001. This work was supported by the Singapore-MIT Alliance, by DARPA and ONR under Grant F49620-01-1-0458, by DARPA and AFOSR under Grant N00014-01-1-0523 (Subcontract 340-6218-3), and by NASA under Grant NAG-1-1978.


Yvon Maday
email: maday@ann.jussieu.fr

Anthony T Patera
email: patera@mit.edu

Gabriel Turinici
email: gabriel.turinici@inria.fr

A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations

We consider "Lagrangian" reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced-basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.

Introduction

The development of computational methods that permit rapid and reliable evaluation of the solution of partial differential equations in the limit of many queries is relevant within many design, optimization, control, and characterization contexts. One particular approach is the reduced-basis method, first introduced in the late 1970s for nonlinear structural analysis [START_REF] Almroth | Automatic choice of global shape functions in structural analysis[END_REF][START_REF] Noor | Reduced basis technique for nonlinear analysis of structures[END_REF], and subsequently developed more broadly in the 1980s and 1990s [START_REF] Barrett | On the reduced basis method[END_REF][START_REF] Fink | On the error behaviour of the reduced basis technique for nonlinear finite element approximations[END_REF][START_REF] Peterson | The reduced basis method for incompressible viscous flow calculations[END_REF][START_REF] Rheinboldt | On the theory and error estimation of the reduced basis method for multi-parameter problems[END_REF][START_REF] Balmes | Parametric Families of Reduced Finite Element Models. Theory and Applications[END_REF]. The reduced-basis method recognizes that the field variable is not, in fact, some arbitrary member of the infinite-dimensional space associated with the partial differential equation; rather, it resides, or "evolves," on a much lower-dimensional manifold induced by the parametric dependence.

Let Y be an Hilbert space with inner product and norm (• , •) Y and

• Y = (• , •) 1/2
Y , respectively. Consider a parametrized "bilinear" form a : Y × Y × D → R, where D ≡ [0, µ max ], and a bounded linear form f : Y → R. We introduce the problem to be solved: Given µ ∈ D, find u ∈ Y such that

a(u(µ), v; µ) = f (v), ∀ v ∈ Y . (1)
Under natural conditions on the bilinear form a (e.g. continuity and coercivity) it is readily shown that this problem admits a unique solution.

We introduce an approximation index N , the parameter sample S N = {α 1 , . . . , α N }, and the solutions u(α k ), k = 1, . . . , N , of problem (1) for this set of parameters. We next define the reduced-basis approximation space

W N = span {u(α k ), k = 1, . . . , N }.
Our reduced-basis approximation is then:

Given µ ∈ D, find u N (µ) ∈ W N such that a(u N (µ), v; µ) = f (v), ∀ v ∈ W N . (2)
This discrete problem is well posed as well under the same former continuity and coercivity conditions. The reduced-basis approach, as earlier developped, is typically local in parameter space in both practice and theory. To wit, the α k are chosen in the vicinity of a particular parameter point µ * and the associated a priori convergence theory relies on asymptotic arguments in sufficiently small neighborhoods of µ * [START_REF] Fink | On the error behaviour of the reduced basis technique for nonlinear finite element approximations[END_REF]. In this paper we present, for single-parameter symmetric coercive elliptic partial differential equations, a first theoretical a priori convergence result that demonstrates exponential convergence of reduced-basis approximations uniformly over an extended parameter domain. The proof requires, and thus suggests, a point distribution in parameter space which does, indeed, exhibit superior convergence properties in a variety of numerical tests [START_REF] Veroy | Reduced Basis Methods Applied to Problems in Elasticity: Analysis and Applications[END_REF]. We refer to [START_REF] Matache | Generalized p-FEM in homogenization[END_REF][START_REF] Matache | Homogenization via p-FEM for problems with microstructure[END_REF] for a different analysis within the homogeneization framework.

Problem Formulation

Let us define the parametrized "bilinear" form a :

Y × Y × D → R as a(w, v; µ) ≡ a 0 (w, v) + µa 1 (w, v) , (3)
where the bilinear forms a 0 : Y ×Y → R and a 1 : Y ×Y → R are continuous, symmetric and positive semi-definite; suppose moreover that a 0 is coercive, inducing a (Y -equivalent) norm ||| • ||| 2 = a 0 (• , •). It follows from our assumptions that there exists a real positive constant γ 1 such that

0 ≤ a 1 (v, v) a 0 (v, v) ≤ γ 1 , ∀ v ∈ Y . (4)
For the hypotheses stated above, it is readily demonstrated that the problem (1) has a unique solution.

Many situations may be modeled by our rather simple problem statement (1), [START_REF] Barrett | On the reduced basis method[END_REF]. For example, if we take Y = H 1 0 (Ω) where Ω is a smooth bounded subdomain of R d=2 , and set a 0 (w, v) = Ω ∇w • ∇v, a 1 = Ω wv, we model conduction in thin plates; here µ represents the convective heat transfer coefficient. If we take Y = H 1 0 (Ω) for Ω ⊂ R d=1,2, or 3 , with Ω 1 ⊂ Ω (Ω 1 , Ω bounded and sufficiently regular), and set a 0 = Ω ∇w • ∇v, a 1 = Ω1 ∇w • ∇v, we model variable-property heat transfer; here 1 + µ is the ratio of thermal conductivities in domains Ω\Ω 1 and Ω 1 . Other choices of a 0 and a 1 can model variable rectilinear geometry, variable orthotropic properties, and variable Robin boundary conditions.

The space Y is typically of infinite dimension so u(µ) is, in general, not exactly calculable. In order to construct our reduced-basis space W N , we must therefore replace u(µ) ∈ Y by a "truth approximation" u N (µ) ∈ Y N ⊂ Y , where u N is the Galerkin approximation satisfying

a(u N (µ), v; µ) = f (v), ∀ v ∈ Y N .
Here Y N , of finite (but typically very high) dimension N , is a sufficiently rich approximation subspace such that |||u(µ)-u N (µ)||| is sufficiently small for all µ in D; for example, for Y = H 1 0 (Ω) we know that, for any desired ε > 0, we can indeed construct a finite-element approximation space,

Y N (ε) , such that |||u(µ) -u N (ε) (µ)||| ≤ ε. It shall prove convenient in what follows to introduce a generalized eigenvalue problem: Find (ϕ N i ∈ Y N , λ N i ∈ R), i = 1, . . . , N , satisfying a 1 (ϕ N i , v) = λ N i a 0 (ϕ N i , v), ∀ v ∈ Y N .
We shall order the (perforce real, non-negative) eigenvalues as 0

≤ λ N N ≤ λ N N -1 ≤ • • • ≤ λ N 1 ≤ γ 1 ,
where the last inequality follows directly from (4). We may choose our eigenfunctions such that

a 0 (ϕ N i , ϕ N j ) = δ i j , (5)
and hence a 1 (ϕ N i , ϕ N j ) = λ N i δ i j , where δ i j is the Kronecker-delta symbol; and such that Y N can be expressed as span {ϕ i , i = 1, . . . , N }. Note that, thanks to the finite dimension of our approximation space Y N , we preclude (the complications associated with) a continuous spectrum -and, as we shall see, at no loss in rigor.

We conclude this section by noting that u N (µ) can be expressed as

u N (µ) = N i=1 f N i ϕ N i 1 + µλ N i , (6)
where

f N i = f (ϕ N i ).

A Priori Convergence Theory

We propose here to choose the sample points α k , k = 1, . . . , N , logequidistributed in D, in the sense that, if we set δ N = ln(γµ max + 1)/N , and γ is any finite upper bound for γ 11 , then

α k = exp{-ln γ + k ℓ=1 δℓN } -γ -1 ,
where we assume that there exists a constant c * such that δkN

δ N ≤ c * , ∀k, k = 1, . . . , N
and also that N ℓ=1 δℓN = ln(γµ max + 1). Denote the reduced-basis approximation space as W N N = span {u N (α k ), k = 1, . . . , N }. Although in general dim(W N N ) ≤ N , we can suppose that dim(W N N ) = N (otherwise we eliminate elements from W N N until it contains only linearly independent vectors). Then, the (reduced basis) problem is :

Given µ ∈ D, find u N N (µ) ∈ W N N such that a(u N N (µ), v; µ) = f (v), ∀ v ∈ W N N . (7)
This problem admits a unique solution.

Our goal is to (sharply) bound |||u N (µ)u N N (µ)|||, for all µ ∈ D, as a function of N (and ultimately N as well). This error bound in the energy norm can be readily translated into error bounds on continuouslinear-functional outputs [START_REF] Prud | Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods[END_REF]; we do not consider this extension further here.

We shall need two standard results from the theory of Galerkin approximation of symmetric coercive problems [START_REF] Strang | An Analysis of the Finite Element Method[END_REF]:

a(u N -u N N , u N -u N N ; µ) = inf w N N ∈W N N a(u N -w N N , u N -w N N ; µ) ; (8)
and

a(u N , u N ; µ) ≤ a(u, u; µ) . (9)
From the definition of the ||| • ||| norm, the positive semidefiniteness of a 1 , (3), ( 4) and ( 8) we can write

|||u N (µ) -u N N (µ)||| 2 ≤ a(u N (µ) -u N N (µ), u N (µ) -u N N (µ), µ) ≤ inf w N N ∈W N N a(u N (µ) -w N N , u N (µ) -w N N , µ) ≤ (1 + µ max γ 1 ) inf w N N ∈W N N |||u N (µ) -w N N ||| 2 , ∀ µ ∈ D. (10)
Also from the definition of the ||| • ||| norm and the positive semidefiniteness of a 1 , (3), ( 4) and ( 9), we obtain

|||u N (µ)||| ≤ (1 + µ max γ 1 ) 1/2 |||u(µ)|||, ∀ µ ∈ D (11)
We begin with a preparatory result in

Lemma 3.1 Let g(z, λ) = 1 1 -λ γ + λe z (12)
for z ∈ Z ≡ [ln(γ -1 ), ∞] and λ ∈ Λ ≡ [0, γ] (recall γ is our strictly positive upper bound for γ 1 ). Then, for any q ≥ 0

|D q 1 g(z, λ)| ≤ 2 q q! , ∀ z ∈ Z, ∀ λ ∈ Λ ,
where D q 1 g denotes the q th -derivative of g with respect to the first argument. Proof. We first remark that for any p ≥ 0

0 ≤ g p (z, λ) ≤ 1, ∀ z ∈ Z, ∀ λ ∈ Λ, , (13)
where g p is the p th -power of g. This follows since ∀ z ∈ Z, e z ≥ γ -1 , and hence,

∀ z ∈ Z, ∀ λ ∈ Λ, 1 -λ/γ + λe z ≥ 1.
We next claim that, for m ≥ 2,

D m-1 1 g(z, λ) = m n=1 a m n g n (z, λ) , (14)
where, for m ≥ 1 (and a 1 1 ≡ 1)

a m+1 1 = -a m 1 a m+1 n = -n a m n + (n -1) 1 -λ γ a m n-1 , 2 ≤ n ≤ m a m+1 m+1 = m 1 -λ γ a m m . (15)
We prove this result by induction. We first differentiate g to obtain

D 1 1 g(z, λ) = -λe z 1 -λ γ + λe z 2 = -1 -λ γ + λe z + 1 -λ γ 1 -λ γ + λe z 2 = -g(z, λ) + 1 - λ γ g 2 (z, λ) ; (16)
(14) and ( 15) for m = 2 directly follows. We now differentiate ( 14) for m = m, and exploit [START_REF] Matache | Generalized p-FEM in homogenization[END_REF], to obtain

D (m+1)-1 1 (z, λ) = m n=1 a m n n g n-1 (z, λ) D 1 1 g(z, λ) = m n=1 a m n n g n-1 (z, λ) -g(z, λ) + 1 - λ γ g 2 (z, λ) = m n=1 -n a m n g n (z, λ) + m+1 n=2 (n -1) 1 - λ γ a m n-1 g n (z, λ) ; (14)
and ( 15) for m = m + 1 directly follows. Thus, ( 14), ( 15) are valid for m ≥ 2, as required. It now follows from ( 13) and ( 14) that, for any q ≥ 1, |D q 1 g(z, λ)| ≤ S q , where S q = q+1 n=1 |a q+1 n |. We now invoke [START_REF] Veroy | Reduced Basis Methods Applied to Problems in Elasticity: Analysis and Applications[END_REF], and observe that ∀ λ ∈ Λ, |1 -λ γ | ≤ 1, to obtain S q ≤ 2q S q-1 ; since S 1 ≤ 2, we conclude that S q ≤ 2 q q!, and the lemma is thus proven.

We now prove a bound for the best approximation result in

Lemma 3.2 For N ≥ N crit inf w N N ∈W N N |||u N (µ) -w N N ||| ≤ |||u N (0)||| exp -N N crit , ∀ µ ∈ D ,
where N crit ≡ c * e ln(γ µ max + 1).

Proof. To facilitate the proof, we shall effect a change of coordinates in parameter space. To wit, we let D ≡ [ln γ -1 , ln(µ max +γ -1 )], and introduce τ : D → D, as τ (μ) = e μγ -1 so that τ -1 (µ) = ln(µ + γ -1 ). We then set ũ(μ) = u(τ (μ)), ũN (μ) = u N (τ (μ)), and ũN

N (μ) = u N N (τ (μ)). We note that ũN (μ) = N i=1 f N i ϕ N i 1 - λ N i γ + λ N i e μ = N i=1 f N i ϕ N i g(μ, λ N i ), (17)
from ( 6), our change of variable, and the definition [START_REF] Prud | Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods[END_REF]. We now observe that in our mapped coordinate, the sample points αk ≡ τ -1 (α k ), k = 1, . . . , N , are equi-distributed with separation αk+1αk ≃ ln(γµ max + 1)/N . It thus follows that, given any μ ∈ D, we can construct a closed interval I μ ∆ of length ∆, that includes μ and M μ( ∆, δ N ) distinct points αP μ n , n = 1, . . . , M . Here M μ( ∆, δ N ) is of the order of ∆ δ N ; more precisely,

M μ( ∆, δ N ) ≥ ∆ c * δ N . ( 18
)
In what follows, we shall often abbreviate M μ( ∆, δ N ) as M . Now, for any μ ∈ D, we introduce ûμ ∈ W N N given by ûμ

≡ M n=1 Q μ n (μ) u N (τ (α P μ n )) = M n=1 Q μ n (μ) ũN (α P μ n ) = M n=1 Q μ n (μ) N i=1 f N i ϕ N i g(α P μ n , λ N i ) ,
where the characteristic functions Q μ n are uniquely determined by

Q μ n ∈ P M -1 ( I μ ∆), n = 1, . . . , M , and Q μ n (α P μ n ′ ) = δ nn ′ , 1 ≤ n, n ′ ≤ M ; here P M -1 ( I μ ∆)
refers to the space of polynomials of degree ≤ M -1 over I μ ∆.

We thus obtain ûμ =

N i=1 f N i ϕ N i [ I μ M -1 g(•, λ N i )] (μ) , (19)
where, for given λ, I μ M -1 g(•, λ) is the (M -1) th -order polynomial interpolant of g(•, λ) through the αP μ n , n = 1, . . . , M ; more precisely,

I μ M -1 g(•, λ) ∈ P M -1 ( I μ ∆), and ( I μ M -1 g(•, λ))(α P μ n ) = g(α P μ n , λ), n = 1, . . . , M . Note that [ I μ M -1 g(•, λ)](τ -1 (µ)) is not a polynomial in µ.
It now follows from ( 5), ( 6), ( 17) and (19) that

|||ũ N (μ) -ûμ ||| ≤ N i=1 f N i ϕ N i g(μ, λ N i ) -[ I μ M -1 g(•, λ N i )] (μ) ≤ sup λ∈Λ |g(μ, λ) -[ I μ M -1 g(•, λ)] (μ)| |||u N (0)||| . (20)
We next invoke the standard polynomial interpolation remainder formula [START_REF] Dahlquist | Numerical Methods[END_REF] and Lemma 3.1 to obtain

sup λ∈Λ |g(μ, λ) -[ I û M -1 g(•, λ)] (μ)| ≤ sup λ∈Λ sup z∈Z 1 M ! |D M 1 g(z, λ)| ∆M ≤ (2 ∆) M μ( ∆,δ N ) . (21)
We now assume that c * δ N 2 ≤ ∆ and ∆ ≤ 1 2 ; under these conditions (recall (18)) we obtain (2 ∆) M μ( ∆,δ N ) ≤ (2 ∆) ∆/c * δ N , and hence, from (20) and (21), we can write

|||ũ N (μ) -ûμ ||| ≤ |||u N (0)|||(2 ∆) ∆/c * δ N . (22)
It remains to select a best ∆ satisfying

c * δ N 2 ≤ ∆ ≤ 1 2 .
To provide the sharpest possible bound, we choose ∆ = ∆ * ≡ 1 2e , the minimizer (over all positive ∆) of (2 ∆) ∆/δ N . Our conditions on ∆ are readily verified: c * δ N 2 ≤ ∆ * follows directly from the hypothesis of our lemma, N ≥ N crit ; and ∆ * ≤ 1 2 follows from inspection. We now insert ∆ = ∆ * into (22) to obtain

|||ũ N (μ) -ûμ ||| ≤ |||u N (0)||| e -N/Ncrit , ∀ μ ∈ D .
It immediately follows that, for any µ ∈ D,

inf w N N ∈W N N |||u N (µ) -w N N ||| = inf w N N ∈W N N |||ũ N (τ -1 (µ)) -w N N ||| ≤ |||ũ N (τ -1 (µ)) -ûτ -1 (µ) ||| ≤ |||u N (0)||| e -N/Ncrit , since û• ∈ W N N and, for µ ∈ D, τ -1 (µ) ∈ D.
This concludes the proof. Then, from [START_REF] Peterson | The reduced basis method for incompressible viscous flow calculations[END_REF], [START_REF] Porsching | Estimation of the error in the reduced basis method solution of nonlinear equations[END_REF], Lemma 3.1, and Lemma 3.2, we obtain Theorem 3.3 For N ≥ N crit ≡ c * e ln(γµ max + 1),

|||u N (µ) -u N N (µ)||| ≤ (1 + µ max γ 1 ) 1/2 |||u N (0)||| e -N/Ncrit , ∀ µ ∈ D; furthermore, |||u(µ) -u N (ε) N (µ)||| ≤ ε + (1 + µ max γ 1 ) |||u(0)||| e -N/Ncrit , ∀ µ ∈ D, for N (ε) such that |||u(µ) -u N (ε) (µ)||| ≤ ε.

Conclusions

We make several observations about the results of Theorem 3.3. First, we obtain exponential convergence with respect to N . Second, our convergence result applies uniformly for all µ ∈ D. Third, our convergence parameter N crit depends only very weakly -logarithmically -on γ 1 , related to the form of the operator, and on µ max , related to the range of the parameter (note we may also view the product γ 1 µ max as the continuitycoercivity ratio). As a result, N crit will, in general, be small, and we will thus achieve convergence "soon" (N ≥ N crit ) with a "large" ( 1 Ncrit ) exponential decay rate. Fourth, we obtain convergence with respect to both N and N : u N N (µ) → u(µ) as N, N → ∞. Let us now make several remarks concerning the point distribution. First, the logarithmic point distribution is intimately related to our particular abstract problem, (1), (3), and, relatedly, the parametric dependence of the solution, [START_REF] Machiels | Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems[END_REF]. In brief, for larger values of µ, the derivatives of (1 + µλ) -1 will be smaller, thus permitting a larger interval in which to recruit the points required for accurate interpolation. Second, it should be clear from the proof of Lemma 3.2 that the requirement on the point distribution is, in fact, rather weak: the location of the M points within I μ ∆ is (save for conditioning issues) irrelevant. This permits, for example, log-random distributions -particularly attractive in higher parameter dimensions in which tensor-product grids are prohibitively costly.

Third, the logarithmic point distribution is not an artifact of our (interpolant-based) proof: in numerical tests [START_REF] Veroy | Reduced Basis Methods Applied to Problems in Elasticity: Analysis and Applications[END_REF] the error in the actual Galerkin approximation is also "minimized" by a logarithmic point distribution; even point distributions that enjoy general optimality properties, such as Chebyshev, do not perform as well as the logarithmic distribution for our particular problem. (Indeed, for Chebyshev interpolation over the interval D, it may be shown that N crit scales as √ µ max -much worse than our ln µ max .)

Fourth, we note that numerical tests [START_REF] Veroy | Reduced Basis Methods Applied to Problems in Elasticity: Analysis and Applications[END_REF] roughly confirm the dependence of |||u N (µ)u N N (µ)||| on N , γ, and µ max . However, in general, our theoretical bound can be quite pessimistic, as might be expected given that our proof is based on (albeit, tailored) interpolation arguments: Galerkin optimality can always do better, for example, choosing to "illuminate" only an appropriate subsample of S N so as to construct the best "subapproximation" (or sub-interpolant) amongst all O(N !) possibilities. This property is no doubt crucial in higher parameter dimensions, in which effective scattered-data higher-order interpolants are very difficult to construct; it is here that the superiority of the reduced-basis approach over simple parameter-space interpolation is most evident. We do not yet have any (uniform in µ) theory for higher parameter dimensions, although numerical results again suggest extremely rapid convergence.

Finally, we note that we address in this paper only one aspect -rapid uniform convergence -of successful reduced-basis approaches. In other papers [START_REF] Machiels | Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems[END_REF][START_REF] Maday | Blackbox reducedbasis output bound methods for shape optimization[END_REF][START_REF] Maday | A blackbox reduced-basis output bound method for noncoercive linear problems[END_REF][START_REF] Prud | Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods[END_REF] we focus on (i) off-line/on-line computational decompositions that permit real-time response, and (ii) a posteriori error estimators that ensure both efficiency and certainty.

Note that γ 1 , γ, and hence S N , are independent of N .

Acknowledgements. We would like to thank Christophe Prud'homme, Dimitrios Rovas, and Karen Veroy of MIT for sharing their numerical results prior to publication. This work was performed while ATP was an Invited Professor at the University of Paris VI in February, 2001. This work was supported by the

Singapore-MIT Alliance, by DARPA and ONR under Grant F49620-01-1-0458, by DARPA and AFOSR under Grant N00014-01-1-0523 (Subcontract 340-6218-3), and by NASA under Grant NAG- 1-1978.