Local Fourier Bases and Modulation Spaces

Turk J Math 30 (2006) , 447 – 462. c TUB¨ ITAK˙

Salti Samarah, Rania Salman

Abstract It is shown that local Fourier bases are unconditional bases for modulation spaces. We prove ﬁrst a version of the Schur test for double sequence with mixed norm and then use it to show boundedness of the analysis operator on the modulation space w Mp,q

Key Words: local Fourier bases, Schur test, mixed norm space, atomic decomposition

1. Introduction

It is an important theme that “nice” functions are unconditional bases for some function spaces. Results of this type are proven in several articles and in almost every book on wavelets. In all those places, the results depend on the particular space and w the bases. In this paper we consider the spaces of modulation spaces Mp,q and the local Fourier bases to be defined later. In [16], Feichtinger, Gröchenig and Walnut proved that the Wilson bases of Daubbechies, Jaffard and Journé constructed in [5] are unconditional bases for modulation spaces. Since the Wilson bases are a special case of the local Fourier bases, we proved in [18] during our work in nonlinear approximation problem that local Fourier bases are unconditional w bases for the modulation spaces Mp,p. In this paper we generalize this result to the case w 6 Mp,q with p = q. Let us start with some definitions and notations. If w is a weight function then the p,q R2 weighted mixed-norm space Lw ( ) is defined to be the space of all measurable functions on R2 for which the following norm is finite:

447 SAMARAH, SALMAN

! Z Z q/p 1/q

k k p,q | |p p f Lw = f(x, y) w(x, y) dx dy , (1) R R with the obvious modiﬁcations if p or q = ∞, in which case we use the supremum. p,q R2 p R2 If p = q,thenLw ( )coincideswithLw ( ). It is known [3] that these spaces are Banach spaces for p, q ≥ 1 and quasi-Banach spaces for 0

i∈Λ1 j∈Λ2

0 × p,q × If w is the restriction of w to the set Λ1 Λ2,then`w0 (Λ1 Λ2)is

0 { k k p,q k k ∞} c =(cij)i∈Λ1,j∈Λ2 : c ` 0 = cw p,q < . w

The H¨older inequality for double sequences with discrete mixed-norm is given as follows 0 0 ∈ p,q ∈ p ,q 0 0 ∞ 1 1 1 1 If a `w and b `1/w , 1

X ≤|| || p,q || || 0 0 aαβbαβ a `w b p ,q . (2) `1/w Z2

The Minkowski’s inequality is given by     p 1/p 1/p X X X X   ≤  | |p aαβ aαβ (3) β α α β for 1 ≤ p<∞ These inequalities will be used later.

2. The Modulation Spaces

In this section we collect some facts about the short-time Fourier transform (STFT) such as its deﬁnition and a pointwise estimation. We also give some results on modulation

448 SAMARAH, SALMAN spaces needed for the subject. Let S be the Schwartz space of smooth functions with rapid decay, and let S0 be the space of tempered distributions. Also let

2πiyt Txf(t)=f(t − x)andMyf(t)=e f(t)(4) be the operators of translation and modulation by x, y ∈ R. Then the STFT of f ∈S0 with respect to the window g ∈Sis deﬁned to be Z −2πiyt Sg f(x, y)=hf, MyTxgi = f(t)g(t − x)e dt (5) R for all x, y ∈ R. In this paper we consider the weighted mixed norm of the function

Sg f(x, y) which induces a norm on f and obtain the so-called modulation spaces. These spaces are a mathematical tool to measure the joint time-frequency distribution of tempered distribution, and they are deﬁned by the decay properties of the STFT. To deﬁne these spaces we consider moderate weights, i.e. non-negative continuous functions satis- fying

w(x + y) ≤ C(1 + |x|)aw(y)forx, y ∈ R, (6)

≥ w ≤∞ for some constants C>0,a 0. The modulation space Mp,q for 0

∞ w is ﬁnite. For p or q = , we use the supremum. If p = q we write Mp ,andifw is constant then we write Mp,q . These spaces were deﬁned by Feichtinger and most of their w ≤ ≤∞ properties were established (see [7, 10, 11, 15]). Mp,q are Banach spaces for 1 p, q and quasi Banach spaces for 0 dual space of Mp,q is the space Mp0 ,q0 where p + p0 =1 1 1 and q + q0 =1 Examples of modulation spaces:

1. Feichtinger’s Segal algebra S0(R)=M1,1 [8].

2 2. M2,2(R)=L (R).

449 SAMARAH, SALMAN

s w | |2 s/2 3. The Bessel potential space H = M2 ,wherew(x, y)=(1+ y ) .

The modulation spaces have nice atomic decomposition. Given ϕ ∈S there exist β>0 and γ>0 small enough and a dual window ϕ◦ ∈S, independent of 1 ≤ p, q ≤∞and w such that every f ∈S0 has an expansion X ◦ f = hf, TβmMγnϕ iTβmMγnϕ. (8) m,n∈Z

∈ w Moreover, f Mp,q if and only if X X q/p 1/q ◦ p p |hf, TβmMγnϕ i| w(βm, γn) < ∞ (9) m∈Z n∈Z

w ≤ ∞ and the sequence space norm in ( 9) is equivalent to the Mp,q-norm. If 1 p, q < , w then the so-called Gabor expansion (8) converges unconditionally in the norm of Mp,q. (see [7, 13, 17]). Next we mention a pointwise estimate of STFT of elements of the set C deﬁned by

N (k) C = C(M, K, N)={g ∈ C (R) : supp g ⊆ [−K, K], max kg k1 ≤ M}, (10) k=0,1,...,N

Lemma 1 [18, 19] Let ϕ ∈ C∞(R),suppϕ ⊆ [−L, L],andC = K + L.Then

1 | |≤ − ∈ R sup Sϕg(x, y) C0 N χ[ C,C](x) for all x, y , (11) g∈C (1 + |y|) with a constant C0 > 0 depending only on M, K, N.

3. Local Fourier Bases

One way to study a signal is to focus on its local properties. In mathematical language this means that if we are given a function f on R, we divide R into intervals Ij =[αj,αj+1] −∞ ∞ with < ... < αj <αj+1 < ... < and study χIj f. An example of a complete 2 orthonormal system for L [αj,αj+1]is s 2 2k +1 π − ϕj,k(x)= χIj (x)sin (x αj), |Ij| 2 |Ij|

450 SAMARAH, SALMAN where k ranges through the non-negative integers Z+.Ifj ranges through the integers Z, then we obtain a basis for L2(R). Such systems are appropriate for focusing on local properties but not for global properties due to the abrupt ”cutoff” effected by multiplication by the characteristic function χIj . In [4], Coifman and Meyer introduced orthonormal bases of this type that involve an arbitrary smooth cut off. In [1] Auscher,

Weiss and Wickerhauser constructed such bases by introducing the bell function bIj using projections such that the set (s ) 2 2k +1 π − ∈ Z ∈ N bIj (x)sin (x αj ),j ,k αj+1 − αj 2 αj+1 − αj is an orthonormal basis for L2(R). Other forms are also given in [1]. The main property N of such bases is for any N ∈ N ∪{∞} there exists a bell function bj ∈ C (R)with 2 support in [αj − j,αj+1 + j+1] such that these sets are orthonormal bases for L (R), where k ≥ >0. If ∆k = αk+1 − αk and {k : k ∈ Z} is the accompanying sequence, then these formes are written [18] as 1 2 1/2 − Tαk M l M l gk, (12) 2 ∆k 2∆k 2∆k or 1/2 1 2 Tα M l− 1 M l− 1 gk, (13) 2 ∆ k 2 − 2 k 2∆k 2∆k where My and Tx are as in (4) and supp gk ⊆ [−k,αk+1 − αk + k+1]. In what follows, we work with the structure (12) since the other is similar, and it − 1 requires only replacing l by l 2 .

4. Unconditional bases For Modulation Spaces

In this section we prove that local Fourier bases are unconditional bases for modulation w spaces Mp,q.

Deﬁnition 1 .Aset{ei,i ∈ I} of vectors in a Banach space B is an unconditional basis, if

451 SAMARAH, SALMAN

(1) the ﬁnite linear combinations of the ei’s are dense in B,andif (2) there exists a constant C ≥ 1, such that X X k ciλieik≤C sup |λi|k cieik i i∈F i∈F holds for any ﬁnite subset F ⊆ I and any sequence (λi) ⊆ CI .

N Theorem 1 Suppose that {ψkl, (k, l) ∈ Z × N}⊆C (R) is a local Fourier base whose 1 ≤ − ≤ underlying partition satisﬁes A αk+1 αk A, A > 1, and infk k = >0.Ifw is a R2 q ≤ ∞ { } weight function on for N>p + a, 0

w with unconditional convergence in the norm of Mp,q. Moreover,   ! 1/q X X q/p 1  p l p  k k w ≤ |h i| ≤ k k w f Mp,q f, ψkl w(αk, ) C f Mp,q (15) C 2∆k k∈Z l∈N for some constant C>0.Ifp = q = ∞,then{ψkl} is a weak basis, that is, the expansion ∗ 1/w (14) converges only in the weak -topology with respect to the predual M1,1 .

Since the Wilson bases are a special case of local Fourier bases [1], then this theorem covers the results given in [16]. the following two corollaries.

Corollary 1 The Wilson basis of exponential decay constructed in [5] is an unconditional w ≤ ∞ basis for Mp,q for 0

Corollary 2 (see [18]) Any local Fourier basis in CN (R) is an unconditional bases for s S0,ifN>1 and H if s

To prove Theorem 1 we use the pointwise estimate for the STFT given in Lemma 1. We consider the set of windows given by

N (k) C = C(M, K, N)={g ∈ C (R):sup g ⊆ [−K, K], max kg k1 ≤ M}. k=0,1,...,N

452 SAMARAH, SALMAN

The explicit construction of the bell functions of a local Fourier basis leads to the following consequence.

1 ≤ − ≤ ≥ ∈ Z Lemma 2 [18] If A αk+1 αk A and if k >0 for all k ,then

{ ∈ Z}⊆C gk = T−αk bk,k (M, K, N) for some M, K, and N.

To prove Theorem 1 so we shall extend the orthonormal expansion (14) from L2(R)to w the modulation spaces Mp,q. For this purpose we study the action of certain associated operators. The analysis operator τ is deﬁned by

τf =(hf, ψkli)(k,l)∈I . (16)

2 2 Since {ψkl, (k, l) ∈ I} is an orthonormal basis, τ is well deﬁned and maps L (R)onto` (I). ∗ The formal adjoint is the synthesis operator τ which acts on “sequences” c =(ckl)(k,l)∈I as X ∗ τ ((ckl)k,l∈I )= cklψkl. (17) (k,l)∈I

The next two propositions show that both operators extend to other function or sequence spaces. We write

l 2 ηkl =(αk, ), (k, l) ∈ Z , 2∆k for the points in the time-frequency plane associated to ψkl, and for a given weight 0 0 function w, w denotes its restriction w (k, l)=w(ηkl) to the discrete set {ηkl}.Then p,q `w0 (I) consists of all sequences on I for which X X q/p 1/q p p kckp,q,w0 = |ckl| w(ηkl) < ∞ . k l

We ﬁrst prove an estimate for the STFT of a local Fourier bases

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Lemma 3 [18, 19] Using the notation of Lemma 1, set

1 G(x, y)=χ − (x) . [ C,C] (1 + |y|)N

N If {ψkl, (k, l) ∈ I}⊆C (R) is a local Fourier bases which satisﬁes the assumptions of

Theorem 1, then there exists C1 > 0, such that

| |≤ ∈ R Sϕψkl(x, y) C1(Tηkl G(x, y)+Tηk,−l G(x, y)) for all x, y . (18)

To prove that τ is bounded, we prove ﬁrst a weighted version of Schur’s Test.

5. A Schur Test For Discrete Weighted Mixed-Norm Spaces

In this section, we prove a version of Schur’s test for discrete weighted mixed-norm p,q spaces `w . Precisely, we prove that if A =(a(i,j),(k,l))i∈I,j∈J,k∈K,l∈L is an inﬁnite matrix, then under some certain conditions the map A from ` p,q to ` p,q is bounded. w1 w2

Lemma 4 Suppose that w1(k, l) and w2(i, j) are two weight functions on the index sets

K × L and I × J respectively, and let A =(a(i,j),(k,l))i∈I,j∈J,k∈K,l∈L be an inﬁnite matrix such that X X −1 −1 |a(i,j),(k,l)|w1(k, l) ≤ C◦w2(i, j) < ∞ for all i ∈ I,j ∈ J (19) k∈K l∈L and ! X X p/q q/p q/p |a(i,j),(k,l)| w2(i, j) ≤ C1w1(k, l)ϕ(k) < ∞ for all k ∈ K, l ∈ L (20) i∈I J∈J

for some constants C◦,C1 > 0 and some sequence ϕ(k) ∈ `u.If1 ≤ p ≤ q<∞ and q p,q × p,q × u = q−p , then the map A is bounded from `w1 (K L) into `w2 (I J).

Proof. The case p = q is treated in [18], so we study the case p =6 q.Letc =(ckl)be p,q 0 1 1 q ∞ 0 k k p,q an element in `w1 ,andlet1

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H¨older inequality (2), viz

! p q/p X X X X k kq p Ac p,q = a(i,j),(k,l)ckl w2(i, j) `w2 j i k l ( ! ) X X X X p q/p p ≤ w2(i, j) a(i,j),(k,l)ckl j i k l ( ! ) p q/p X X X X 1 1 + 0 1 − 1 p p p 0 0 = w2(i, j) a(i,j),(k,l) |ckl|w1(k, l) p p j i k l ( ! X X X X p p p 0 ≤ w2(i, j) a(i,j),(k,l) |ckl| w1(k, l) p j i k l  ! p q/p 0 X X p  · −1 a(i,j),(k,l) w1(k, l)  k l

( ) q X X p X X p 0 p p p p − 0 p 0 ≤ w2(i, j) C◦ w2(i, j) p a(i,j),(k,l) |ckl| w1(k, l) p j i k l ( ) q/p q X X X X 0 p p p 0 = C◦ w2(i, j) a(i,j),(k,l) |ckl| w1(k, l) p j i k l

where we used H¨older inequality (2) in the third inequality and condition (19) in the q fourth inequality. Now we write kAck p,q in the form `w2

 ( )  q/p p/q p/q 0 X X X X p q p/p 0 k k ≤  | |p p  Ac p,q C◦ a(i,j),(k,l) w2(i, j) ckl w1(k, l) . `w2 j k l i

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Using Minkowski’s inequality (3) twice, we get

 ( )  q/p p/q 0 X X X X p p p/p  p 0  kAck p,q ≤ C◦ a(i,j),(k,l) w2(i, j)|ckl| w1(k, l) p `w2 k j l i   n o p/q 0 X X X X p q/p p/p  p 0  ≤ C◦ a(i,j),(k,l) w2(i, j)|ckl| w1(k, l) p k l i j   p/q 0 X X p X X p/p p 0  q/p q/p = C◦ |ckl| w1(k, l) p a(i,j),(k,l) w2(i, j) . k l i j

Now, using condition (20), we have

0 X X p p p/p p 0 +1 kAck p,q ≤ C◦ C1 |ckl| w1(k, l) p ϕ(k) `w2 k l

0 X X p/p p p = C◦ C1 |ckl| w1(k, l) ϕ(k). k l

p q−p Since q + q = 1, we apply H¨older inequality and get   ! p/q ! q−p q/p q 0 X X X p p/p q k k ≤  | |p p  · q−p Ac p,q C◦ C1 c(k,l) w1(k, l) ϕ(k) . `w2 k l k

Therefore 0 1/p 1/p 1/p kAck p,q ≤ C◦ C kck p,q kϕ(k)k < ∞ `w2 1 `w1 `u 2

The boundedness of the analysis operator is the content of the following proposition.

w Proposition 1 . Under the hypotheses of Theorem 1, τ is a bounded operator from Mp,q p,q ≤ ≤∞ into `w0 (I) for 1 p, q .

The proof of this proposition is based the Schur test as well as the following lemma

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P Lemma 5 Let φ(k)=( (1 + C + |γn − l |)aq/p(1 + | l − γn|)−Nq/p)p/q , l 2∆k 2∆k q 1 ≥ q ≤ ≤ ∈ and N>p + a, A < ∆k

p q−p X X q−p q k k | − l | aq/p | l − | −Nq/p φ `u = (1 + C + γn ) (1 + γn ) 2∆k 2∆k k l

l l − = (1 + C + |γn − |)a(1 + | − γn|) N 2∆k 2∆k u0,u

l l − ≤ (1 + C + |γn − |)a(1 + | − γn|) N 2∆k 2∆k 1,u

q q−p X X − l l − q p q = (1 + C + |γn − |)a(1 + | − γn|) N 2∆k 2∆k k l

X l l − = (1 + C + |γn − |)a(1 + | − γn|) N 2∆k 2∆k l `u

X l l − ≤ (1 + C + |γn − |)a(1 + | − γn|) N 2∆k 2∆k l `1 X X l l − = (1 + C + |γn − |)a(1 + | − γn|) N . 2∆k 2∆k k l

The last equality is less than X l l − sup (1 + C + |γn − |)a(1 + | − γn|) N , 2∆k 2∆k k l

q 1 ≥ (see [18]) and since N>p + a>1+a, A < ∆k

k k ∞ φ `u < , which implies that. φ ∈ `u 2

Now we prove proposition 1.

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Proof. The cases p = q =1andp = q = ∞ are treated in [18, 19]. For 1 ≤ p ≤ q<∞, we use lemma 4. Given ϕ ∈ C∞ with compact support and β, γ small enough, there exists [17] a dual window ϕ◦ ∈S, such that every f ∈S0 has an atomic decomposition X ◦ f = hf, TβmMγnϕ iTβmMγnϕ, m,n∈Z

∈ w with f Mp,q, if and only if

X X q/p 1/q ◦ p p |hf, TβmMγnϕ i| w(βm, γn) < ∞. m∈Z n∈Z

Then X ◦ (τf)kl = hf, TβmMγnϕ ihTβmMγnϕ, ψkli . m,n

Therefore the proposition is proved if we can show that the map deﬁned by the matrix

A(k,l),(m,n) = hTβmMγnϕ, ψkli maps the sequence

h ◦i∈ p,q Z2 cmn = f, TβmMγnϕ `w1 ( ),w1(m, n)=w(αm, γn)

p,q 0 into `w0 (I),w(k, l)=w(ηkl). For this it is enough to verify the conditions of Schur’s test. Since the ﬁrst condition is similar to the ﬁrst condition in [18], it is enough to verify the second condition of Schur’s test. We write (6) in the form l l a w(αk, ) ≤ w(βm, γn) 1+|αk − βm| + | − γn| . (21) 2∆k 2∆k

Now, using this inequality, we estimate the sum

X X p/q q/p l q/p Σ= |hTβmMγnϕ, ψkli| w(αk, ) 2∆k k∈N l∈Z

458 SAMARAH, SALMAN as X X q/p q/p Σ≤ |hTβmMγnϕ, ψkli| w(βm, γn) k∈Z l∈Z l aq/p p/q · 1+|αk − βm| + | − γn| 2∆k X X q/p l l ≤C1w(βm, γn) G(βm − αk,γn− )+G(βm − αk,γn+ ) 2∆k 2∆k k∈Z l∈Z l aq/p p/q · 1+|αk − βm| + | − γn| 2∆k X X aq/p l l ≤C1w(βm, γn) 2sup{G(βm−αk,γn− ),G(βm−αk ,γn+ )} k 2∆k 2∆k k∈Z l∈Z l aq/p p/q · 1+|αk − βm| + | − γn| . 2∆k

Since |βm − αk|≤C and l l −N G(βm − αk,γn− ) ≤ χ[−C,C](βm − αk) · (1 + | − γn|) , 2∆k 2∆k we have X X p/q a+1 a l aq/p l −Nq/p Σ ≤ 2 C1C w(βm, γn) (1 + C + |γn − |) (1 + | − γn|) . 2∆k 2∆k k∈Z l∈Z

1 ≤ − ≤ Since A αk+1 αk A, A > 1, we have X X p/q a+1 a l aq/p l −Nq/p Σ ≤ 2 C1C w(βm, γn) (1 + C + |γn − |) (1 + | − γn|) . 2∆k 2∆k k l P Let φ(k)=( (1 + C + |γn − l |)aq/p(1 + | l − γn|)−Nq/p)p/q. Then by using l 2∆k 2∆k lemma 5 we conclude that (φ(k))) is in `u. and we have the second condition of lemma 4. 2 w w 6 The following proposition is proved in [18] for Mp,q ,p= q,WeproveitforMp,q ,p= q for convenience.

459 SAMARAH, SALMAN

∗ p,q w ≤ ≤ ≤∞ Proposition 2 The map τ isaboundedmapfrom`w0 into Mp,q for 1 p q .

Proof. Assume that p, q < ∞,andletc =(ckl)(k,l)∈I be ﬁnitely supported. If ∈ 1/w w h Mp0 ,q0 , the dual of Mp,q, then proposition 1 implies

X X h i | |≤k k k k 0 0 ≤k k k k k k cklψkl,h = ckl(τh)kl c p,q (τh) p ,q c p,q τ op h 1/w ` 0 ` 0 ` 0 M 0 0 w 1/w w p ,q k,l k,l

Hence the estimates X ∗ k k w k w | h i| ≤ k k p,q k k τ c M = M sup cklψkl,h c ` 0 τ op p,q p,q w khk 1/w ≤1 M 0 0 k,l p ,q

∗ p,q k k show that τ is bounded on `w0 where . op is the operator norm. Furthermore, for any >0 there exists a ﬁnite subset F ⊆ I, such that   ! 1/q X X X q/p  p p  k k w ≤k k | | cklψkl Mp,q τ op ckl w(ηkl) < (k,l) k l

∗ where (k, l) ∈/ F , for all ﬁnite subsets F ⊇ F. This means that τ c converges unconditionally. ∞ ∞ 1/w ∗ If p = or q = , then taking the supremum over M1 shows that τ is bounded w ∗ 2 on M∞,∞ and that sum is w -convergent. Proof of Theorem 1: ∗ w p,q Since τ and τ are bounded on Mp,q and `w0 ,theidentity X ∗ f = hf, ψkliψkl = τ τf (22) k,l

2 w ∞ extends from L (IR )toMp,q ,1

∗ ∗ k k w ≤k k k h i k p,q ≤k k k k k k w f M τ op ( f, ψkl )(k,l)∈I ` 0 τ op τ op f M . p,q w p,q P Furthermore, since in a (ﬁnite) linear combination f = k,l cklψkl the coeﬃcients are uniquely determined as

ckl = hf, ψkli =(τf)kl,

460 SAMARAH, SALMAN we can estimate X ∗ k k w k k w λklcklψkl Mp,q = τ (λklckl) Mp,q k,l ∗ ≤kk k k p,q τ op (λklckl)(k,l) ` 0 w ∗ ≤kk k k k k p,q τ op λ ∞ c ` 0 w ∗ ≤kk k k∞kk k k k w τ op λ τ op f Mp,q , where λ =(λk,l)(k,l)∈I and c =(ckl)(k,l)∈I . This shows that {ψkl, (k, l) ∈ I} is an w unconditional basis for Mp,q . Therefore, Theorem 1 is proved in a general case. (See [18]) 2

Remark 1 We can get all result in [18] if we take p = q, and φ(k)=1, Note that the conditions in lemma 4 becomes exact conditions in Schur test in [18]; and proposition (1, 2) becomes proposition 1 , 2 in [18].

References

[1] Auscher, P.: Remarks on the local Fourier bases. In Wavelets: Mathematics and Applica- tions, J. Benedetto and M. Frazier, eds., pp. 203 – 218, CRC Press, Boca Raton, 1994.

[2] Auscher, P., Weiss, G., Wickerhauser, M.: Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets. In Wavelets: A Tutorial in Theory and Applications, C. K. Chui, ed., pp. 237-256, Academic Press, San Diego, 1992.

[3] Benedeck, A., Panzone, R.: The spaces Lp with mixed norm, Duke. Math. J. 23, 301-324 (1961).

[4] Coifman, R., Meyer, Y.: Remarques sur l’analyse de Fourier à fenètre, sèrie I, C. R. Acad. Sci. Paris 312, 259-261 (1991).

[5] Daubechies, I., Jaﬀard, S., Journ´e, J.: A simple Wilson orthonormal basis with exponential decay. SIAM J. Math. Anal. 22, 554-572 (1991).

[6] Donoho, D.: Unconditional bases are optimal bases for data compression and for statistical estimation, Applied and Computation Harmonic Analysis 1, 100-115 (1993).

[7] Feichtinger, H.G.: Atomic characterizations of modulation spaces through Gabor-type representations, Proc. Conf ”Constructive Function Theory”, Edmonton, July 1986, Rocky Mount. J. Math. 19, 113-126 (1989).

[8] Feichtinger, H.G.: On a new Segal algebra, Monatsh. Math. 92, 269-289 (1981).

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[9] Feichtinger, H.G.: Banach convolution algebras of Wiener type. In Proc. Conf. Functions, Series, Operators, Budapest, Colloquia Math. Soc. J. Bolyai, pages 509-524, Amsterdam- Oxford-New York, North Holland 1980.

[10] Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. Technical report, University of Vienna 1983.

[11] Feichtinger, H.G., Gr¨ochenig, K.: Banach spaces related to integrable group representations and their atomic decompositions, I, J. Func. Anal. 86, 307-340 (1989).

[12] Feichtinger, H.G., Gr¨ochenig, K.: Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math. 108, 129-148 (1989).

[13] Feichtinger, H.G., Gr¨ochenig, K.: Non-orthogonal Wavelet and Gabor Expansions, and Group Representations. In ”Wavelets and their Applications”, Ruskai, M.B., et al., eds, pp. 353-376, Jones and Bartlett, Boston,

[14] Feichtinger, H.G., Gr¨ochenig, K.: Gabor Wavelets and the Heisenberg Group: Gabor expansions and Short Time Fourier Transform from the Group Theoretical Point of View, In “Wavelets- A Tutorial in Theory and Applications”, C. K. Chui, ed., Academic Press, (1992).

[15] Feichtinger, H.G., Gr¨ochenig, K.: A uniﬁed approach to atomic characterization via integrable group representations. In Proc. Conf. Lund June 1986, lect. Notes in Math. 1302, pages 53-73 (1988). MR 89h 46035 (M. Milman, introduction), 1988.

[16] Feichtinger, H.G.: K. Gr¨ochenig, D.Walnut, Wilson basis and modulation spaces. Math. Nachr. 155, 7-17 (1992).

[17] Gr¨ochenig, K.: Describing Functions : Atomic decompositions versus frames, Monatsh. Math. 112, 1-42 (1991).

[18] Samarah, S.: Modulation Spaces and Non-linear Approximation. Ph.D thesis, University of Connecticut, (1998).

[19] Gr¨ochenig, K., Samarah, S.: Nonlinear approximation with local Fourier bases, Const. Approx. 16, 317-331 (2000).

w [20] Galperin, Y.V., Samarah, S.: Time-frequency analysis and modulation spaces Mp,q Appl. Comput. Harmon. Anal. 16, 1-18 (2004).

Salti SAMARAH, Rania SALMAN Received 10.05.2005 Department of Mathematics, Jordan University of Science & Technology, Irbid-JORDAN e-mail: [email protected]

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