1. Introduction The α-modulation spaces M s,α

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1 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES LASSE BORUP AND MORTEN NIELSEN Abstract. The α-modulation spaces M p,q (R d ), α [,1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed, and as an application we consider Jackson-type estimates for m-term nonlinear approximation. 1. Introduction The α-modulation spaces M p,q (R d ), α [, 1], are a parameterized family of spaces that include the Besov and modulation spaces as special cases corresponding to α = 1 and α =, respectively. Besov spaces, see e.g. [21] for the definition, are based on coverings of frequency space R d by balls B(a n, r n ) satisfying a n B(a n, r n ) 1/d, that is to say there exist constants c, C (, ) such that c a n B(a n, r n ) 1/d C a n for all the balls. On the other hand, the modulation spaces introduced by Feichtinger in [7] are based on uniform coverings of the frequency space, i.e., coverings satisfying a n B(a n, r n ) 1/d, and it was pointed out by Feichtinger and Gröbner [8, 6] that Besov and modulation spaces are special cases of an abstract construction, the so-called decomposition type Banach spaces D(Q, B, Y ). Gröbner [11] used the methods in [8] to define the α-modulation spaces as a family of intermediate spaces. Gröbners idea was to define spaces corresponding to coverings based on the rule a n α B(a n, r n ) 1/d, α 1. The precise definition of an α-modulation space will be given in Section 2. The coverings giving rise to α-modulation spaces have also been considered (independently) by Päivärinta and Somersalo in [19]. Päivärinta and Somersalo used the partitions to extend the Calderón-Vaillancourt boundedness result for pseudodifferential operators to the local Hardy spaces h p. The family of α-modulation spaces arise naturally in several applications. In [1], pseudodifferential operators on α-modulation spaces are studied in the univariate case. It was proved that certain pseudodifferential operators with exotic symbols of Hörmander type extends naturally to bounded operators on α-modulation spaces. The proof are based on the brushlet characterization of the α-modulation spaces given in [3]. The mapping properties of pseudodifferential operators on α-modulation spaces are also studied by Holschneider and Nazaret in [18]. These results can be seen as extensions of earlier classical results by Córdoba and Fefferman [5]. Pseudodifferential operators on modulation spaces have also been studied, see e.g. [2, 13]. One successful approach to study function spaces and operators on such spaces is to construct an unconditional basis for the space and use the corresponding norm characterization of the elements in the space to study various operators on the space. One striking example is the study of Calderón-Zygmund operators in smooth wavelet bases, see e.g. [16]. Another family of orthonormal bases for L 2 (R) is brushlet bases. Brushlets are the Key words and phrases. Modulation spaces, α-modulation spaces, Banach Frames. This work is in part supported by the Danish Technical Science Foundation, Grant no

2 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 2 image of a local trigonometric basis under the Fourier transform, and such systems were introduced by Laeng [14]. Later Coifman and Meyer [15] used brushlets as a tool for image compression. The present authors proved in [2] that nice brushlets form unconditional bases for L p (R d ), 1 < p <. In [3], the freedom to choose the frequency localization of a brushlet system was used to construct (orthonormal) unconditional brushlet bases for the univariate α-modulation spaces. The structure of the real line was used extensively in [3], and we cannot see any straightforward way of extending the orthonormal bases to the multivariate setting. In the present paper we consider Banach frames for α-modulation spaces. Banach frames, introduced by Gröchenig in [12], is a (much) weaker concept than an unconditional basis but they still provide stable expansions and a discrete characterization of the norm in the Banach space. Fornasier has studied Gabor-type Banach frames for univariate α- modulation spaces in [9]. However, as Fornasier also states in [9], it also seems to be technically difficult to extend the construction to the multivariate case. Here we introduce an easy construction of Banach frames based on brushlet systems for multivariate α- modulation spaces. The frames constructed are not orthonormal bases, but each system is locally orthonormal (in a certain sense that will be explained in Section 3). We construct an explicit dual system to the frames with essentially the same structure as the frame itself. As an application of the frames, we consider Jackson-type estimates for m-term nonlinear approximation. The structure of the paper is as follows. In Section 2, we give the precise definition of the α-modulation spaces. Section 3 contains the construction of the multivariate brushlet systems that will form frames for the α-modulation spaces. We also give a discrete characterization of the norm in the α-modulation spaces using the frame constructed, this is done in Section 3.2. We recall the definition of a Banach frame in Section 3.3. A stable reconstruction operator is explicitely constructed using a suitable (non-canonical) dual frame which is defined in Section 4. In Section 4 we can finally conclude that the frames constructed are indeed Banach frames for the α-modulation spaces. In Section 5 we consider an application of the Banach frames to approximation theory and derive Jackson estimates for best m-term approximation. Finally, there is an appendix where we prove some technical results related to the partitions of unity used to define the α-modulation spaces. 2. α-modulation spaces We now define the α-modulation spaces. The spaces are defined by a parameter α, belonging to the interval [, 1]. This parameter determines a segmentation of the frequency domain from which the spaces are built. Thus, we need to define nice partitions of the frequency space. Let B(c, r) R d denote the ball with center c and radius r, and let x := (1 + x 2 ) 1/2. Definition 2.1. A countable set Q of subsets Q R d is called an admissible covering if R d = Q Q Q and there exists n < such that #{Q Q : Q Q } n for all Q Q. Let r Q = sup{r R: B(c r, r) Q for some c r R d }, R Q = inf{r R: Q B(c R, R) for some c R R d } denote respectively the radius of the inscribed and circumscribed sphere of Q Q. An admissible covering is called an α-covering, α 1, of R d if Q x αd (uniformly) for all x Q and for all Q Q, and there exists a constant K 1 such that R Q /r Q K for all Q Q. We also need partitions of unity compatible with the covers from Definition 2.1. We let F(f)(ξ) := (2π) d/2 R d f(x)e ix ξ dx, f L 2 (R d ), denote the Fourier transform.

3 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 3 Definition 2.2. Given p (, ] and an α-covering Q of R d. A corresponding bounded admissible partition of unity of order p (p-bapu) {ψ Q } Q Q is a family of functions satisfying supp(ψ Q ) Q Q Q ψ Q(ξ) = 1 sup Q Q 1/ p 1 F 1 ψ Q L p <, p := min(1, p). Remark 2.3. It is proved in Lemma A.1 that an α-covering with a corresponding p-bapu actually exist for every α [, 1] and p >. For Q Q define the multiplier ψ Q (D)f := F 1 (ψ Q Ff), f L 2 (R d ). By [21, Proposition 1.5.1], the conditions in Definition 2.2 ensure that ψ Q (D) extends to a bounded operator on L p (R d ), < p, uniformly in Q Q. We have the following definition of the α-modulation spaces. Definition 2.4. Given < p, q, s R, and α 1, let Q be an α-covering of R d for which there exists a p-bapu Ψ. Then we define the α-modulation space, Mp,q (R d ) as the set of distributions f S (R d ) satisfying (2.1) f M p,q := ( Q Q ξ Q qs ψ Q (D)f q L p ) 1/q <, with {ξ Q } Q Q a sequence satisfying ξ Q Q. For q = we have the usual change of the sum to sup over Q Q. It is easy to see that (2.1) defines a quasi-norm (or a norm if p, q 1) on Mp,q (R d ) and that two different sequences {ξ Q } Q Q give equivalent norms. Furthermore, Theorem 3.1 below, shows that two different p-bapu s give equivalent norms too. Thus, the α- modulation space is well defined. Notice that the definition is given for the full range of p and q, extending Gröbners original definition in [11]. It can be proved that Mp,q (R d ) is a quasi-banach space, and that S(R d ) Mp,q (R d ) S(R d ), see [3]. Moreover, if p, q <, S(R d ) is dense in Mp,q (R d ) Admissible coverings. In this section we discuss a specific construction of an α- covering of R d. This type of covering was considered in [11] and in [19]. A proof of Lemma 2.5 below can be found in [11], but since Gröbner s work has never been published, we have included a proof for the sake of completeness. Construction of α-coverings are also considered (from another perspective) in [19]. Notice that the set of balls {B(k, d)} k Z d \{} is an admissible -covering of R d. Define for some β ( 1, ), the bijection δ β on R d by δ β (ξ) := ξ ξ β (with inverse δ β, β = β/(1 + β)). Since the set {B(k, R)} k Z d \{} is admissible for R d, so is {δ β (B(k, R))} k Z d \{}. Moreover, we have the following result. Lemma 2.5. Suppose β. Given R >, there exists an r >, such that (2.2) δ β (B(z, R)) B(δ β (z), r z β ), for all z R d, with z 1. Likewise, given r > there exists an R >, such that (2.3) B(δ β (z), r z β ) δ β (B(z, R)), for all z R d.

4 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 4 Proof. The proof is based on the following observation. For two points x, z R d and β ( 1, ), we have δ β (x) δ β (z) = x x β z z β x x β x z β x z + β z z β = x x β z β + z β x z ( (2.4) = β x x β 1 + z β) x z for some x L(x, z), by the mean value theorem. Given R >, suppose x B(z, R). Then since z 1, (2.4) yields δ β (x) δ β (z) r z β for some r > depending only on β and R. Now, take any y δ β (B(z, R)), i.e., y = δ β (x) for some x B(z, R). Then y δ β (z) r z β, which proves (2.2). We turn to (2.3). Suppose first that z K for some K > r 1+β. Then it is easy to verify that there exists a radius P > such that B(δ β (z), r z β ) B(, P) for all z. Likewise, there exists a radius R such that B(, P) δ β (B(z, R)) for all z. This proves (2.3) for z K. Suppose now that z > r 1+β. Recall that δ 1 β = δ β, where β := β/(β + 1). Thus, to show the inclusion (2.3) is equivalent to show that (2.5) δ β (B(z, r z β )) B(δ β (z), R). Suppose x B(z, r z β ) for some β > 1, then (1 r z (1+β ) ) z x (1 + r z (1+β ) ) z. Since 1 + β = (1 + β ) 1, (2.4) yields δ β (x) δ β (z) R z β z β = R for some R > depending only on r and β. Now, take any y δ β (B(z, r z β )), i.e., y = δ β (x) for some x B(z, r z β ). Then, y δ β (z) R, which proves (2.5). We can now deduce the following result from Lemma 2.5. Theorem 2.6. Given α < 1, let β = α/(1 α). Then there exists a constant r 1 > such that (2.6) {B(k k β, r k β )} k Z d \{} is an α-covering for any r > r 1. Proof. By (2.2) there exists a radius r 1 such that R d k Z d \{}B(δ β (k), r k β ) for all r r 1. Fix such an r and let R := R(r) be given such that (2.3) holds. Then, since {δ β (B(k, R(r)))} k Z d \{} is an admissible covering of R d, so is {B(δ β (k), r k β )} k Z d \{}. It is easy to see that B(δ β (k), r k β ) y dα for all y B(δ β (k), r k β ) independent of k Z d \ {}. Thus {B(δ β (k), r k β )} k Z d \{} is an α-covering for any r > r 1. Denote by Q(c, r) the cube with center c and side lengths 2r. We have the following corollary to Theorem 2.6. Corollary 2.7. Given α < 1, let β = α/(1 α). Then there exists a constant r 1 > such that is an α-covering of R d for any r > r 1. {Q(k k β, r k β )} k Z d \{}

5 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES A specific α-covering. Given α < 1. In Section 3.2 below we will give an equivalent norm for the α-modulation spaces using a fixed α-covering P defined as follows. Let r 1 be the constant from Theorem 2.6 and fix r 3 > r 2 > r 1. Then according to Theorem 2.6 and Corollary 2.7 the sets (2.7) and (2.8) O := O α,r := {B(k k α/(1 α), r 2 k α/(1 α) )} k Z d \{} P := P α,r := {Q(k k α/(1 α), r 3 k α/(1 α) )} k Z d \{} are α-coverings. In the following sections we will use the short notation B k := B(k k α/(1 α), r 2 k α/(1 α) ) and Q k := Q(k k α/(1 α), r 3 k α/(1 α) ). 3. Multivariate brushlet systems Univariate brushlet bases have proven successfull in characterizing univariate α-modulation spaces. However, it is still an open question how to construct orthonormal multivariate brushlet bases. In this section we define separable multivariate brushlet systems and study their localization properties. The study concludes in Section 4 where we will see that nice separable brushlet systems in fact constitutes Banach Frames for multivariate α-modulation spaces Brushlet systems. Let us first recall the definition of a univariate brushlet. Take a non-negative ramp function ρ C r (R), for some r 1, satisfying { for ξ, (3.1) ρ(ξ) = 1 for ξ 1. Given < ε < 1 r 2 /r 3, define g by ( ) ξ (3.2) ĝ(ξ) := ρ ρ ε where ĝ denotes the Fourier transform of g. For an interval I = [a I, a I ( 1 ξ ), we define the bell function (3.3) b I (ξ) := ĝ ( I 1 (ξ a I ) ) ( ξ ai = ρ ε I ε ), ) ρ ( a I ξ Notice that supp(b I ) I and b I (ξ) = 1 for ξ [a I + ε I, a I ε I ]. Now for each n N we define the univariate brushlet w n,i by ( 2 (3.4) ŵ n,i (ξ) = I b I(ξ) cos π ( ) n + 1 )ξ a I 2. I The brushlets also have an explicit representation in the time domain. Let for notational convenience ) e n,i := π( n I Then, I (3.5) w n,i (x) = 2 eia Ix { g ( I (x + e n,i ) ) + g ( I (x e n,i ) )}. By a straight forward calculation it can be verified that there exists a constant C <, such that (3.6) g(x) C(1 + εx ) r, ε I ).

6 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 6 with r 1 given by the smoothness of the ramp function. Thus a univariate brushlet w n,i essentially consists of two humps at ±e n,i. We call r in (3.6) the decay rate of the brushlet. Let Q = d i=1 I i be a cube in R d, and let w n,q := d w ni,i i, n N d, i=1 be the associated multivariate brushlet. Notice that {w n,q } n N d is an orthonormal system in L 2 (R d ) for a fixed cube Q. We say that the brushlet w n,q has decay rate r > if each w ni,i i, i = 1, 2,...,d, has decay rate r. We associate a family of projection operators to the brushlets as follows. These operators will be used to obtain an equivalent norm for the α-modulation spaces in Section 3.2. Given an interval I R, define the operator P I : L 2 (R) L 2 (R) by P I f(ξ) := b I (ξ) [ b I (ξ) ˆf(ξ) + b I (2a I ξ) ˆf(2a I ξ) b I (2a I ξ) ˆf(2a I ξ) ]. It can be verified that P I is an orthogonal projection, mapping L 2 (R) onto Span{w n,i : n N }. For Q = d i=1 I i a cube in R d, we define the operator P Q by the corresponding tensor product. Clearly, P Q is a projection operator P Q : L 2 (R d ) Span{w n,q : n N d }. Moreover, given f L 2 (R d ) we have supp( P Q f) Q, and P Q f(ξ) = ˆf(ξ) for all cubes Q = Q(c, r) provided ξ Q(c, (1 ε)r). Finally, notice that, [ d ] (3.7) P Q = S Q (Id i +R aii R a Ii ) S Q, i=1 where ŜQf := b Q ˆf and Ra f(x) := e i2a f( x), x, a R Characterization of α-modulation spaces. Now we show that it is possible to express the Mp,q (R d )-norm using the projection operators P Q associated with the α- covering P. This leads to a characterization of the α-modulation space norm using the multivariate brushlet system. The main result is the following. Theorem 3.1. Given α < 1, < p, q, and s R. Let P be the disjoint α-covering defined in (2.8), and let P Q, Q P, be the associated projection operators generated from a brushlet system with decay rate r > max(1, 1/p). Then for any f Mp,q (R d ) we have ( ) 1/q. (3.8) f M ξ p,q Q qs P Q f q L p Q P Proof. Let Ψ be a p-bapu subordinate to an α-covering Q. Take f M p,q (R d ). Then, P Q f = Q A Q P Q (ψ Q (D)f), Q P, in S (R), where A Q is the set of cubes Q Q with Q Q. According to Lemma B.2 in Appendix B, sup #A Q d A <. Q P

7 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 7 Using (3.3) we have that F 1 b Q Lp C p Q 1 1/p, for < p. Now, by the identity (3.7) and since ψ Q (D)f L p, < p, for any Q Q, Proposition in [21] implies P Q f Lp C Q 1/ p 1 F 1 b Q L p ( p := min(1, p)) C Q A Q ψ Q (D)f Lp Q A Q ψ Q (D)f Lp with C independent of Q. Clearly, ξ Q ξ Q for any ξ Q Q, ξ Q Q, when Q A Q. Hence (3.9) ξ Q s P Q f Lp C Q A Q ξ Q s ψ Q (D)f Lp for < p, with C independent of Q. Suppose < q 1, then ( ) q ( ) q ξ Q s ψ Q (D)f Lp = 1 AQ (Q ) ξ Q s ψ Q (D)f Lp Q P Q A Q Q P Q Q ( 1AQ (Q ) ξ Q s ) q, ψ Q (D)f Lp Q Q Q P where 1 AQ (Q ) = 1 for Q A Q and for Q Q \ A I. Since 1 AQ (Q ) = 1 AQ (Q), for any Q P and Q Q, this gives ( ) q ξ Q s ψ Q (D)f Lp da ξ Q qs ψ Q (D)f q L p. Q P Q A Q Q P Likewise, for q =, sup ξ Q s ψ Q (D)f Lp d A sup sup ξ Q s ψ Q (D)f Lp Q P Q A Q P Q A Q Q = d A sup ξ Q s ψ Q (D)f Lp. Q Q For 1 < q <, Hölder s inequality with 1 = 1/q + 1/q implies ( Q P ) q ξ Q s ψ Q (D)f Lp Q A Q ( (1 AQ (Q )) q ) q/q ( ( 1AQ (Q ) ξ Q s ) ) q ψ Q (D)f Lp Q P Q Q Q P d q 1 A 1 AQ (Q ) ( ξ Q s ) q ψ Q (D)f Lp Q P Q Q d A ξ Q qs ψ Q (D)f q L p. Q Q The lower bound in (3.8) now follows by combining the above estimates with the inequality (3.9). The upper bound can be proved in a similar fashion. We now have a characterization of the M p,q (R d )-norm using the Lebesgue norm of the projection operators P Q. But this Lebesgue norm can be given by the size of the brushlet coefficients.

8 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 8 Corollary 3.2. Suppose f L 2 (R d ), and Q is a cube in R d. Given p (, ]. If each brushlet w n,q, n N d, has decay rate r > max(1, 1/p), then P Qf L p (R d ) if and only if { f, w n,q } n N d l p. In fact, if one of these conditions are satisfied we have ( (3.1) P Q f Lp Q p f, w n,q p) 1/p, < p <, with equivalence independent of Q. When p = the sum in (3.1) is changed to sup over n N d. Proof. From (3.6) we have that g L p (R). This, together with the representation (3.5), imply that (3.11) sup x R d w n,q (x) p C Q p 2 and sup w n,q p L p C Q p 2 1. Suppose p 1. Let c be the center of the cube Q. Since F(P Q fw n,q ) is compactly supported, we have (see e.g. [21, p. 18]) f, w n,q p = P Q f, w n,q p ( e ix c P Q f(x)w n,q (x) ) p dx n N d n N d R d C Q 1 p P Q f(x) p w n,q (x) p dx C Q 1 p 2 PQ f p L p. R d Likewise, P Q f p L p n N d f, w n,q p w n,q p L p C Q p 2 1 f, w n,q p. For 1 < p < the lemma follows using the two estimates (3.11) for p = 1, together with Hölder s inequality (see e.g. [17, 2.5]). The case p = is left for the reader. Let us introduce a new and more compact notation for the brushlets w n,q associated with the α-covering P given in (2.8). Recall that each cube Q P is of the form Q = Q k = Q(k k α 1 α, r k α 1 α ), k Z d \ {}. For Q k P we use the short hand notation w n,k := w n,qk, k Z d \ {}. Using Lemma 3.2 we can derive the following result from Theorem 3.1. Proposition 3.3. Given < p, q, and s R. Let B = {w n,k } k Z d \{},n N d be a brushlet system with decay rate r > max(1, 1/p) associated with the α-covering P, α < 1. Then we have the characterization f M p,q ( k Z d \{} ( ( 1 αd (s+ k 1 α 2 αd p ) f, w n,k ) ) p q/p) 1/q. Inspired by Proposition 3.3 let us define the following coefficient space. Definition 3.4. Given < p, q, s R, and α < 1, we define the space m p,q as the set of coefficients c = {c n,k } n N d,k Z d \{} C satisfying c m p,q := ( k Z d \{} ( ( 1 αd (s+ k 1 α 2 αd p ) c n,k ) ) p q/p) 1/q <.

9 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES Banach Frames. With Proposition 3.3 in mind, we recall the definition of a Banach frame in Definition 3.5 below. However, let us first recall the definition of a frame for L 2 (R d ). A countable subset G = {g n : n Z d } L 2 (R d ) is a frame for L 2 (R d ) if there exist constants < A, B < such that A f 2 L 2 n Z d f, g n 2 B f 2 L 2, f L 2 (R d ). Define the coefficient (analysis) operator C = C G by C G f = { f, g n } n Z d, and the synthesis operator D = D G = C G by Dc = n c ng n A Banach frame is defined as follows (see [12]). Definition 3.5. A Banach frame for a separable Banach space B is a sequence G = {g n : n Z d } in the dual space B with an associated sequence space B d on Z d such that the following properties hold. (1) The coefficient operator C G is bounded from B into B d. (2) Norm equivalence: f B { f, g n } n Z d Bd, f X. (3) There exists a bounded operator R from B d onto B, a so-called synthesis or reconstruction operator, such that RC G f = R{ f, g n } n Z d = f. When B = L 2 (R d ) and B d = l 2 (Z d ), Definition 3.5 coincides with the usual definition of a frame for L 2 (R d ). Remark 3.6. Proposition 3.3 shows that the first two requirements of Definition 3.5 are satisfied by our multivariate brushlet system with B = Mp,q and B d = m p,q. All that remains to verify is the existence of a stable reconstruction operator R. This will be done in Section Banach frames for α-modulation spaces We now have the tools needed to show that a nice brushlet system constitute a Banach frame for the α-modulation spaces. First we will define the system of functions that will turn out to be a dual to the brushlet system B = {w n,k } n N d,k Z d \{}. Recall the α- covering O = {B k } k Z d \{} in (2.7). Given the multiplier ψ k (D) := ψ Bk (D), we define the system B := { w n,k } n,k where (4.1) w n,k := ψ k (D)w n,k. We have the following result which shows that (B, B) is a pair of dual frames for L 2 (R d ). Proposition 4.1. Let B = {w n,k } n N d,k Z d \{} be a collection of brushlet functions in L 2 (R d ), with α-covering P given by (2.8), α < 1. Let B = { w n,k } n N d,k Z d \{} be the associated system defined by (4.1). Then (B, B) is a pair of dual frames for L 2 (R d ). Proof. Recall that the covering P has finite height, i.e., k Z d \{} χ Q k (ξ) C < for some fixed constant C. We can therefore write P = N j=1 P j, with {w n,k } n N,Q k P j an orthonormal system for j = 1, 2,...,N. The brushlet system B can thus be written as a union of orthonormal sequences and it follows that B is a Bessel system. Notice that

10 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 1 ψ k (D)P Qk = ψ k (D) since supp(ψ Bk ) B k Q(k k α/(1 α), (1 ε)r 3 k α/(1 α) ). Thus, ψ k (D)f, w n,k 2 = ψ Bk ˆf, ŵn,k 2 = ψ Bk ˆf 2 2 P Qk f 2 2 = ˆf, ŵ n,k 2 = f, w n,k 2, where we used that {w n,k } n N d is an orthonormal system and that ψ Bk 1. It follows that B is a Bessel system. Moreover, ψ k (D)f = P Qk ψ k (D)f = ψ k (D)f, w n,k w n,k = f, w n,k w n,k. Now, {ψ k } k is a partition of unity, so we have f = ψ k (D)f = f, w n,k w n,k, f L 2 (R d ), k Z d \{} k Z d \{} where the last sum converges unconditionally in L 2 (R d ) since B and B are Bessel systems. Hence, (B, B) is a pair of dual frames for L 2 (R d ). We also have the dual representation (see e.g., [4, Lemma 5.6.2]) (4.2) f = f, w n,k w n,k, f L 2 (R d ). k Z d \{} We want to extend the result in Proposition 4.1 to the α-modulation spaces. Thus we need a bounded reconstruction operator R : m p,q Mp,q (R d ). The following Lemma indicate how we can define such an operator. Lemma 4.2. Given < p, q, s R, and α < 1. For any finite sequence {c n,k } n N d,k Z d \{} of complex numbers we have M c n,k w n,k C {c n,k } n,k m. p,q p,q n,k Proof. Let f = n,k c n,k w n,k. We have ( f M = k qs/(1 α) ψ p,q k (D)f q p k k ) 1/q ( ( = k qs/(1 α) ψk (D) k k A k k A k ) q c n,k w n,k ( C k qs/(1 α) c n,k w n,k C ( k qs/(1 α) q c n,k w n,k k n n n p q p ) 1/q, where we have used that O has finite height and that ψ k (D) is a bounded operator on L p (R d ), uniformly in k. It is easy to verify that w n,k satisfies similar inequalities as in ) 1/q p ) 1/q

11 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 11 (3.11). Thus, using the same technique as in the proof of Corollary 3.2 yields α Lp c n,k w n,k C k 1 α ( d 2 d p ) {c n,k } lp, < p, uniformly in k Z d \ {}. Next we show that the dual pair B, B form Banach frames for the α-modulation spaces. We define the coefficient operator C : Mp,q (R d ) m p,q by Cf = { f, w n,k } n,k and the reconstruction operator R : m p,q Mp,q (R d ), by {c n,k } n,k n,k c n,k w n,k. Theorem 4.3. Given < p, q, s R, and α < 1. Suppose the brushlet system has decay rate r > max(1, 1/p). Then the coefficient operator C and reconstruction operator R are both bounded and make Mp,q (R d ) a retract of m p,q(r d ), i.e., RC = Id M p,q (R d ). In particular, for p, q 1, B and B are Banach frames for the α-modulation spaces Mp,q (R d ). Proof. We consider (4.2) in Mp,q (R d ). According to Proposition 3.3, the coefficient operator C is a bounded linear operator, and using Lemma 4.2 it is easy to verify that R is a bounded linear operator. Thus, (4.2) extends to a bounded splitting RC = Id M p,q (R d ) as illustrated in the following commuting diagram. M p,q Id M p,q M p,q C m p,q R 5. Nonlinear approximation In this section we consider an application of the Banach frames for the α-modulation spaces to nonlinear approximation. We are mainly interested in Jackson estimates for best m-term approximation, and using the stabillity of the frames, we can apply a general theory introduced in [1]. We let B = {w n,k } k Z d \{},n N d be a brushlet system with decay rate r >, and associated with an α-covering P for some < α 1. Consider the normalized functions ω n,k = w n,k, k Z d \ {}, n N d w n,k. M p,p (R d ) Notice that for any finite brushlet expansion f = n,k c n,kω n,k we have, ( f M p,p (R d ) C k Z d \{}, c n,k p) 1/p, 1 r < p, by Proposition 3.3, i.e., {ω n,k } n,k is l p -hilbertian (as defined in [1]). Let us introduce some notation that will be needed to explore nonlinear approximation with brushlet bases. Let D = {g k } k N be a dictionary in a quasi-banach space X. We consider the collection of all possible m-term expansions with elements from D: { } Σ m (D) := c i g i ci C, #Λ m. i Λ

12 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 12 The error of the best m-term approximation to an element f X is then σ m (f, D) X := inf f f m X. f m Σ m (D) The corresponding approximation spaces are defined as follows. Definition 5.1 (Approximation spaces). The approximation space A γ q(x, D) is defined by ( ) ( f A γ q (X,D) := m γ ) q 1 1/q σ m (f, D) X <, m m=1 and (quasi)normed by f A γ q (X,D) = f X + f A γ q (X,D) for < q, γ <, with the l q norm replaced by the sup-norm, when q =. We can now use the stability of the brushlet frames in the α-modulation spaces and apply [1, Theorem 6] to obtain a Jackson inequality for the brushlet frames. Proposition 5.2. Let {w n,k } k Z d \{},n N d be a brushlet system with decay rate r 1, associated with a disjoint α-covering P for some α 1. Let D = { } w n,k / w n,k M p,p (R d ) k Z d \{},. Then Mτ,τ β,α (R d ) A γ ( τ M p,p (R d ), D ), provided p > 1 r and γ = 1 τ 1 p = β s αd >. Proof. The Proposition follows from [1, Theorem 6] since {ω n,k } n,k is l p -hilbertian, and one can verify (using the notation of [1]) that Kτ(M τ p,p (R d ), D) = Mτ,τ β,α (R d ) with β = s + αd ( 1 τ 1 p). Appendix A. Bounded admissible partitions of unity and their properties The α-modulation spaces are defined using a specific bounded admissible partition of unity, but the spaces are actually independent of the specific choice. Here we construct a p-bapu for any α and any < p <. The reader can find a more algebraic construction in [8] that works for p 1. We have the following construction. Proposition A.1. For α [, 1), < p, there exists an α-covering of R d with a corresponding p-bapu {ψ k } k Z d \{} S(R d ) satisfying for every multi-index β and k Z d \ {}. β ψ k (ξ) C β ξ β α, Proof. For r >, and k Z d \ {} we define the ball B r k := {ξ Rd : ξ k α 1 α k < r k α 1 α }. By Lemma 2.5, there exists r 1 > such that {B r 1 k } k Z d \{} is an α-covering of R d. There also exists < r 2 < r 1, such that {B r 2 k } k Z d \{} is pairwise disjoint. Fix r > r 1. We take Φ C (R d ) satisfying inf ξ B(,r1 ) Φ(ξ) := c > and supp(φ) B(, r). Let g k (ξ) := Φ( c k α (ξ c k )), k Z d \ {}, where c k := k α 1 α k. Clearly, gk C (R d ) with supp(g k ) Bk r. In fact, {supp(g k)} k is an α-covering of R d. The covering is admissible (see Lemma 2.5) since {B r 2 k } k Z d \{}, with B r 2 k supp(g k), is pairwise disjoint. It is easy to see that the partition has finite height, i.e., k Z d \{} χ supp(g k )(ξ) n 1 for some uniform constant n 1.

13 Notice that BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 13 β g k (ξ) = c k α β ( β Φ)( c k α (ξ c k )) C β c k α β, and since c k 1 for all k Z d \ {}, we have β g k (ξ) C β c k α β ξ α β for all ξ B r k. Since we want a p-bapu, we consider the sum g(ξ) := k Z d \{} g k(ξ). Now, {supp(g k )} k has finite height, so g is well-defined, and the finite overlap ensures that β g(ξ) C β ξ β α. Recall that g k (ξ) c for all ξ B r 1 k, and since {Br 1 k } k Z d \{} covers R d, we have g(ξ) c. Thus, we can define ψ n (ξ) := g n (ξ) k Z d \{} g k(ξ). It is straightforward to show that β ψ k (ξ) C β ξ β α. In order to conclude, we need to verify that sup Q Q 1/ p 1 F 1 ψ k L p <, where p = min{1, p}. Let ψ k (ξ) = ψ k ( c k α ξ + c k ) = Φ(ξ) k Φ( c k α c k α (ξ c k ) + c k ). Notice that for every β N d there exists a constant C β independent of k Z d \ {} such that (A.1) β ξ ψ k (ξ) C β χ B(,r) (ξ). By a simple substitution in each of the following integrals, we obtain F 1 ψ p k = L p ψ k (ξ)e ix ξ p dξ dx R d R d = c k αd( p 1) ψk (ξ)e ix ξ p dξ dx, and since c k dα Q, R d R ( d ) p Q p 1 C d β ψk L1 x R d 1 dx C d Q p 1, d β (d+1)/ p where we have used integration by parts and (A.1) for the last estimate. We conclude that {ψ k } k is a p-bapu corresponding to the α-covering {supp(g k )} k. Appendix B. Some properties of α-coverings Let s d = π d/2 /Γ( d 2 + 1) be the volume of the unit ball in Rd. Given an α-covering Q, let r Q and R Q denote respectively the radius of the inscribed and circumscribed sphere of Q Q. Let K 1 be such that R Q /r Q K for all Q Q. Notice that for Q Q we have s d r d Q Q s d R d Q, so s d Q r d Q = Rd Q r d Q Q R d Q K d s d, and consequently Q RQ d rd Q independent of Q. Given two α-coverings Q and Q, suppose Q Q and Q Q have nonempty intersection. Then from the observation above, we have R Q R Q. Let d Q and d Q denote the center of the circumscribed sphere of Q and Q respectively, and let c Q be the center of the inscribed sphere of Q. Then there exists a constant κ > 2K such that (B.1) Q B(d Q, R Q ) B(d Q, κ 2K R Q) B(c Q, κr Q ). Lemma B.1. Given an α-covering Q, there exist n < subsets Q i Q, i = 1, 2,...,n, such that Q = n i=1 Q i and the elements of Q i are pairwise disjoint.

14 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 14 The proof is given in Lemma c.1.1 and Lemma c.8.3 in [11], but will be given here for completeness. Proof. Given Q Q define Q := {Q Q: Q Q }. By (B.1) we have Q Q Q B(c Q, κr Q ). Since Q has finite height, there exists a constant n 2 such that n 2 C d Q n 2 s d κ d r d Q > Q Q Q c(#q ) Q, i.e., #Q is bounded by a constant n independent of Q. Let Q 1 Q be a maximal set of pairwise disjoint elements, and let inductively Q i Q \ i 1 k=1 Q k, i = 2, 3,..., be a maximal set of pairwise elements. Suppose Q Q n +1. Then for each k = 1, 2,...,n there exists a Q k Q k such that Q k Q. But this is a contradiction to the fact that #Q n. Lemma B.2. Let Q and Q be two α-coverings. For each Q Q let A Q = {Q Q : Q Q }. Then there exists a constant d A such that #A Q d A independent of Q. Proof. Recall that there exists a constant δ such that Q δ Q for all Q A Q, independent of Q Q. According to (B.1) and Lemma B.1 we have i.e., #A Q κd n δs d. Q s d r d Q δs d κ d n (#A Q ) Q, Acknowledgment. The authors would like to thank Hans G. Feichtinger and Massimo Fornasier for their comments which improved this manuscript. References [1] L. Borup. Pseudodifferential operators on α-modulation spaces. J. Funct. Spaces Appl., 2(2):17 123, 24. [2] L. Borup and M. Nielsen. Approximation with brushlet systems. J. Approx. Theory, 123(1):25 51, 23. [3] L. Borup and M. Nielsen. Nonlinear approximation in α-modulation spaces. Preprint, 25. [4] O. Christensen. An introduction to frames and Riesz bases. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston, MA, 23. [5] A. Córdoba and C. Fefferman. Wave packets and Fourier integral operators. Comm. Partial Differential Equations, 3(11):979 15, [6] H. G. Feichtinger. Banach spaces of distributions defined by decomposition methods. II. Math. Nachr., 132:27 237, [7] H. G. Feichtinger. Modulation spaces of locally compact abelian groups. In R. Radha, M. Krishna, and S. Thangavelu, editors, Proc. Internat. Conf. on Wavelets and Applications, pages Allied Publishers, New Delhi, 23. [8] H. G. Feichtinger and P. Gröbner. Banach spaces of distributions defined by decomposition methods. I. Math. Nachr., 123:97 12, [9] M. Fornasier. Banach frames for α-modulation spaces. Preprint, 25. [1] R. Gribonval and M. Nielsen. Nonlinear approximation with dictionaries. I. Direct estimates. J. Fourier Anal. Appl., 1(1):51 71, 24. [11] P. Gröbner. Banachräume glatter Funktionen und Zerlegungsmethoden. PhD thesis, University of Vienna, [12] K. Gröchenig. Describing functions: atomic decompositions versus frames. Monatsh. Math., 112(1):1 42, [13] K. Gröchenig and C. Heil. Modulation spaces and pseudodifferential operators. Integral Equations Operator Theory, 34(4): , [14] E. Laeng. Une base orthonormale de L 2 (R) dont les éléments sont bien localisés dans l espace de phase et leurs supports adaptés à toute partition symétrique de l espace des fréquences. C. R. Acad. Sci. Paris Sér. I Math., 311(11):677 68, 199.

15 BANACH FRAMES FOR MULTIVARIATE α-modulation SPACES 15 [15] F. G. Meyer and R. R. Coifman. Brushlets: a tool for directional image analysis and image compression. Appl. Comput. Harmon. Anal., 4(2): , [16] Y. Meyer. Ondelettes et opérateurs. II. Actualités Mathématiques. [Current Mathematical Topics]. Hermann, Paris, 199. Opérateurs de Calderón-Zygmund. [Calderón-Zygmund operators]. [17] Y. Meyer. Wavelets and operators, volume 37 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, [18] B. Nazaret and M. Holschneider. An interpolation family between Gabor and wavelet transformations: application to differential calculus and construction of anisotropic Banach spaces. In Nonlinear hyperbolic equations, spectral theory, and wavelet transformations, volume 145 of Oper. Theory Adv. Appl., pages Birkhäuser, Basel, 23. [19] L. Päivärinta and E. Somersalo. A generalization of the Calderón-Vaillancourt theorem to L p and h p. Math. Nachr., 138: , [2] K. Tachizawa. The boundedness of pseudodifferential operators on modulation spaces. Math. Nachr., 168: , [21] H. Triebel. Theory of function spaces, volume 78 of Monographs in Mathematics. Birkhäuser Verlag, Basel, Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK- 922 Aalborg East, Denmark address: lasse@math.aau.dk and mnielsen@math.aau.dk