CHARACTERIZATIONS OF THE MULTIVARIATE WAVE PACKET SYSTEMS. Guochang Wu and Dengfeng Li* 1. INTRODUCTION

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1 TIWNESE JOURNL OF MTHEMTICS Vol. 18, No. 5, pp , Octoer 2014 DOI: /tjm This paper is availale online at CHRCTERIZTIONS OF THE MULTIVRITE WVE PCKET SYSTEMS Guochang Wu and Dengfeng Li* stract. Some characterizations of the multivariate wave packet systems in terms of the Fourier transforms of the wave packet systems generating functions are discussed in this paper. First, the characterizations of the orthogonal wave packet systems are provided. Secondly, a sufficient condition of the completeness of wave packet system in some special cases is estalished. Finally, the necessary conditions and sufficient conditions for the wave packet systems to e wave packet Parseval frames with the very general lattices are otained. Thus, the corresponding known results in Gaor systems and wavelet systems are otained as some corollaries. 1. INTRODUCTION Today, we are living in a data world. On the one hand, people have to develop the good ways to process all kinds of data. On the other hand, they are faced with analyzing the accuracy of such methods and providing a deeper understanding of the underlying structures. In the late 18th century, the Fourier Transform ecame the first tool to analyze the data and has achieved the greatest achievements. However, the Fourier Transform has a serious disadvantage: the local perturation of a function leads to a change of all Fourier coefficients simultaneously. This deficiency led to the irth of applied harmonic analysis, which is nowadays one of the major research areas in applied mathematics and engineering. It exploits methods not only from harmonic analysis, ut also from areas such as approximation theory, numerical mathematics and operator theory. Now, applied harmonic analysis plays an important role in engineering such as signal processing, image processing, digital communications, medical imaging, compressed sensing, and so on. Received Decemer 12, 2013, accepted January 23, Communicated y Chin-Cheng Lin Mathematics Suject Classification: 42C15, 42C40. Key words and phrases: Gaor system, Wavelet system, The wave packet system. The work is supported y the National Natural Science Foundation of China (No ) and the Natural Science Foundation of Henan Province (No ). *Corresponding author. 1389

2 1390 Guochang Wu and Dengfeng Li Gaor systems were first introduced y Gaor [1] in 1946, with the Gaussian window for the purpose of constructing efficient, time-frequency localized expansions of finite-energy signals. They are generated y modulations and translations of a finite family of functions. Casazza and Christensen [2] gave some important characterizations aout Gaor frames for suspaces of L 2 (R) in Then, Shi and Chen[3] estalished some new necessary conditions for Gaor frames. These conditions are also sufficient for tight frames. Recently, Li et al. [4] presented some new sufficient conditions for Gaor frame via Fourier transform. Gaor systems can only give the time-frequency content of a signal with a constant frequency and time resolution. This is often not the most desired resolution, which leads to a irth of wavelet analysis. Wavelet systems are otained y shifting and dilating a finite family of functions. They have attracted considerale interests from the mathematical community and from memers of many diverse disciplines since Dauechies and his cooperators [5] comined the theory of the continuous wavelet transform with the theory of frames to define wavelet frames for L 2 (R). In 1990, Dauechies [6] otained the first result on the necessary condition for wavelet frames, and then in 1993, Chui and Shi [7] otained an improved result. In 1978, Cordoa and Fefferman [8] introduced wave packet systems y applying certain collections of dilations, modulations and translations to the Gaussian function in the study of some classes of singular integral operators. In paper [9], authors devoted to descrie any collections of functions which are otained y applying the same operations to a finite family of functions. In fact, Gaor systems and wavelet systems are special cases of wave packet systems. Wave packet systems have recently een successfully applied to prolems in harmonic analysis and operator theory [10, 11] and attracted people s attention. The properties of wave packet systems have een investigated y many authors. For example, in [12], authors studied oth the continuous and discrete versions of wave packet systems y using a unified approach and gave a classification of the wave packet system to e a Parseval frame. They constructed a very general example of wave packet frame. The paper [13] considered wave packet systems as special cases of generalized shift-invariant systems and presented a sufficient condition for a wave packet system to form a frame. nalogues of the notion of Beurling density to descrie completeness properties of wave packet systems via geometric properties of the sets of their parameters is introduced in [14], and the necessary conditions for existence of wave packet frames were otained and the large families of new, non-standard examples of wave packet frames with prescried dimensions were provided. Since oth Gaor systems and wavelet systems are some particular examples of wave packet systems, people ask naturally: how do we construct some examples of wave packet systems such that they possess simultaneously oth Gaor systems and wavelet systems advantages and, however, overcome their shortcomings? In need of

3 Characterizations of the Multivariate Wave Packet Systems 1391 applications, how do we develop the algorithm as classical multiresolution analysis in the setting of the wave packet systems? So far as we know, few results are known aout these prolems. The main goal of this paper is to characterize the multivariate wave packet systems in terms of the Fourier transforms of the wave packet systems generating functions. Some characterizations of all kinds of the orthogonal wave packet systems with different operator order and with aritrary expanding matrix dilations are estalished. lso, a sufficient condition of the completeness of wave packet systems in some special cases is derived. Then, the necessary conditions and sufficient conditions aout the wave packet systems to e wave packet Parseval frames with the very general lattices are presented. Thus, the corresponding known results in Gaor systems and wavelet systems are otained as some corollaries. Of course, our method comines with some techniques in wavelet analysis and time-frequency analysis. We mainly orrow some thoughts in classifying the wavelet frame and Gaor systems in [15,16,17]. Let us now descrie the organization of the paper. Section 2 is of a preliminary character: it contains various notations and some facts aout the frame and the wave packet system. In Section 3, the characterizations of all kinds of the orthogonal wave packet systems with different operator orders are estalished. Finally, the necessary conditions and sufficient conditions aout the wave packet systems to e the Parseval frames with the very general lattices are otained. 2. PRELIMINRIES Some asic notations are listed in this section. Throughout this paper, we use the following notations. R and Z denote the set of real numers and the set of integers, respectively. L 2 (R n ) is the space of all square-integrale functions in n dimensions, and and denote the inner product and norm in L 2 (R n ), respectively, and l 2 (Z n ) denotes the space of all square-summale sequences. We denote y T n the n-dimensional torus. By L p (T n ) we denote the space of all Z n -periodic functions f (i.e., f is 1-periodic in each variale) such that T n f(x) p dx < +. We use the Fourier transform in the form (2.1) ˆf(ω) = R n f(x)e 2πix ω dx, where denotes the standard inner product in R n, and it is often omitted when we can understand this from the content. Sometimes, ˆf (ω) is defined y Ff. Let E n denote the set of all expanding matrices. n expanding matrix means that all of its eigenvalues have magnitude greater than 1. For E n,wedenotey the transpose of. It is ovious that E n. Let GL n (R) denote the set of all n n non-singular matrices with real entries. For B GL n (R) we denote y B 1 the inverse of B. For the sake of simplicity, we denote ( ) 1 y.

4 1392 Guochang Wu and Dengfeng Li Let us recall the definition of frame. Definition 2.1. Let H e a separale Hilert space. sequence {f i } i N of elements of H is a frame for H if there exist constants 0 <C D< such that for all f H, (2.2) C f 2 <f,f i 2 D f 2. i=1 The numers C, D are called lower and upper frame ounds, respectively (the largest C and the smallest D for which (2.2) holds are the optimal frame ounds). Those sequences which satisfy only the upper inequality in (2.2) are called Bessel sequences. frame is tight if C = D. IfC = D =1, it is called a Parseval frame. In this paper, we will work with three unitary operators on L 2 (R n ).Let E n and B, C GL n (R). The first one consists of the dilation operator D : L 2 (R n ) L 2 (R n ) defined y (D )f(x) =q 1 2 f(x) with q = det. The second is the shift operator T Bk : L 2 (R n ) L 2 (R n ),k Z n, defined y (T Bk f)(x) =f(x Bk). The final one consists of the modulation operator E Cm : L 2 (R n ) L 2 (R n ),m Z n, defined y (E Cm f)(x) =e 2iπCm x f(x). Let P Z, Q R n and S = P Q. Then, we have S Z R n. gain, let { p : p P } E n and B GL n (R). For the function ψ L 2 (R n ),thewave packet system Ψ is defined y (2.3) Ψ = { ψ p, ν, m (x) D p E ν T Bm ψ(x), m Z n, (p, ν) S }. It is easy to see that if p = j (j Z) and S = Z {0} in (2.3), then we otain the wavelet systems. Of course, the Gaor systems can e got when the set { p : p P } in (2.3) only consists of the elementary matrix E. This simple oservation suggests that the wave packet systems provide greater flexiility than the wavelet systems or the Gaor systems. By changing the order of the operators, we can also define the following one-to-one function systems from S Z n into L 2 (R n ): (2.4) Ψ 1 = { ψ p, ν, m (x) D p T Bm E ν ψ(x), m Z n, (p, ν) S }, (2.5) Ψ 2 = { ψ p, ν, m (x) E ν D p T Bm ψ(x), m Z n, (p, ν) S }. Then, we will give the definitions of the wave packet frame and the frame wave packet function. Definition 2.2. We say that the wave packet system Ψ defined y (2.3) is a wave packet frame if it is a frame for L 2 (R n ). Then, the function ψ is called a frame wave packet function.

5 Characterizations of the Multivariate Wave Packet Systems 1393 In order to prove results presented in next section, we need some lemmas. t first, we will consider the following set of functions: { (2.6) Γ= f L 2 (R n ): ˆf L (R n ) and ˆf has compact support in R n } \{0}. Lemma 2.1. Γ is a dense suset of L 2 (R n ). The following two lemmas come from the ook [17]. Lemma 2.2. Suppose that {f k } + k=1 is a family of elements in a Hilert space H such that there exist constants 0 <C D<+ satisfying (2.2) for all f elonging to a dense suset D of H. Then, the same inequalities (2.2) are true for all f H; that is, {f k } + k=1 is a frame for H. Lemma 2.3. The system {ψ(x Cm)} is orthogonal if and only if (2.7) ˆψ(ω + C m) 2 = c ψ 2,a.e. ω R n, where c = detc. The following useful fact can e found in [12, Lemma 2.2]. Lemma 2.4. Let GL n (R), y, z R n and f L 2 (R n ). Then the following holds: (1) (T y f)ˆ = E y ˆf, (Ez f)ˆ = T z ˆf, (D f)ˆ= D ˆf; (2) T y E z f = e 2πiz y E z T y f, D E y f = E yd f, D T y f = T 1 yd f; (3) (T y E z f)ˆ= e 2πiz y T z E y ˆf; (4) (D T y f)ˆ(ξ) =E yd ˆf(ξ) = det 1 2 ˆf( ξ)e 2πi 1y ξ. 3. THE CHRCTERIZTION OF THE ORTHOGONL WVE PCKET SYSTEMS In this section, we will characterize the orthogonalityof all kindsof the wave packet systems with different operator order in terms of the Fourier transforms of the wave packet systems generating functions. Therefore, some existing results in Gaor system and wavelet system are otained as some corollaries. For convenience, we only study the special cases of wave packet systems defined y (2.3). Theorem 3.1. Let E n and B GL n (R). Wave packet system {D j E νt Bm ψ(x)} j Z,,ν S is orthogonal if and only if oth of the equation (3.1) ˆψ(ω + B m) 2 = ψ 2,a.e. ω R n

6 1394 Guochang Wu and Dengfeng Li and the equation (3.2) ˆψ( j (ω + B m) ν) ˆψ(ω + B m)=0,j N, a.e. ω R n hold, where = detb and ν 0. Proof. (= ) We firstly assume that wave packet system {D j E νt Bm ψ(x)} j Z,,ν S is orthogonal. In particular, the system {D 0 E 0T Bm ψ(x)} is also orthogonal. By Lemma 2.3, the equation (3.1) holds. Furthermore, when j 1 <j 2, y changing variales, we have (3.3) 0= D j 1 E ν1 T Bm1 ψ( ), D j 2 E ν2 T Bm2 ψ( ) = det j 1 +j 2 2 e 2iπν1j1x ψ( j 1 x Bm 1 )e 2iπν2j2x ψ( j 2 x Bm2 )dx R n = det j 2 j 1 2 ψ(x)e 2iπ(ν 1 ν 2 j2 j1)(x+bm 1 ) R n ψ( j 2 j 1x + B( j 2 j 1m1 m 2 ))dx =e 2iπ(ν 2 j 2 j 1 ν 1 )Bm 1 ψ( ),E ( j 2 j 1ν2 ν 1 ) Dj 2 j 1 T B(m2 j 2 j 1m 1 ) ψ( ). Set C = e 2iπ(ν 2 j 2 j 1 ν 1 )Bm 1, ν = j 2 j 1 ν 2 ν 1,j = j 2 j 1 and m =(m 2 j 2 j 1 m 1 ),thenforj 1 <j 2, (3.4) D j 1 E ν1 T Bm1 ψ( ), D j 2 E ν2 T Bm2 ψ( ) = C ψ( ), E ν D j T Bmψ( ), where j>0 (j N), ν S, ν 0and m Z n. Therefore, the orthogonality etween D j 1 E ν1 T Bm1 ψ(x) and D j 2 E ν2 T Bm2 ψ(x), forj 1 <j 2, ν 1,ν 1 S, m 1,m 2 Z, can e reduced to the orthogonality etween ψ(x) and E ν D j T Bmψ(x),wherej>0 and ν S, ν 0,m Z. It follows from Lemma 2.4 and Plancherel theorem that (3.5) ψ( ), E ν D j T Bmψ( ) = det j 2 ˆψ( j ω ν) ˆψ(ω)e 2πiBmω dω. It follows from Levi theorem that the interchange of the order of summation and integration is valid in the following. Then, we have ˆψ( j (ω + B s) ν) ˆψ(ω + B s) dω (3.6) = ˆψ( j ω ν) ˆψ(ω) dω R n ( ˆψ( j ω ν) 2 dω) 1 2 ( ˆψ(ω) 2 dω) 1 2 <, R n R n R n

7 Characterizations of the Multivariate Wave Packet Systems 1395 where the inequality is otained y using Cauchy-Schwarz s inequality. Thus the two series in (3.1) and (3.2) asolutely converge for a.e. ω R n, consequently, we can define a function F j : R C y (3.7) F j (ω) = ˆψ( j (ω + B s) ν) ˆψ(ω + B s), a.e. ω. It is clear to see that F j (ω) is B T -periodic, and the aove argument gives that F j (ω) L 1 (B [0, 1] n ). In fact, we even have F j (ω) L 2 (B [0, 1] n ) y (3.8) F j (ω) 2 ˆψ( j (ω + B s) ν) 2 ˆψ(ω + B s) 2. Then, it deduce from the definition of F j (ω) that ˆψ( j ω ν) ˆψ(ω)e 2πiBmω dω R n (3.9) = ˆψ( j (ω + B s) ν) ˆψ(ω + B s)e 2πiBmω dω = F j (ω)e 2πiBmω dω. Comining with (3.3)-(3.5) and (3.9), we find (3.10) 0 = F j (ω)e 2πiBmω dω. That is to say that, for any j N, the all Fourier coefficients of the functions F j (ω) are 0. This shows that F j (ω) =0,a.e. ω R n. So the equation (3.2) holds. ( =) Suppose that the equations (3.1) and (3.2) hold. Then it derive from (3.1) and Lemma 2.3 that the system {D 0 E 0T Bm ψ(x)} m Z is orthogonal. For fixed j Z, ν S, since (3.11) D j E νt Bm1 ψ( ), D j E νt Bm2 ψ( ) = D 0 E 0 T Bm1 ψ( ), D 0 E 0 T Bm2 ψ( ) = δ m1,m 2, the system {D j E νt Bm ψ(x)} m Z is also orthogonal. Notice that (3.2) holds, hence according to the proof of (3.4), (3.5) and (3.9), we otain that for j 1 <j 2,j:= j 2 j 1 and a.e. ω R n, (3.12) D j 1 E ν1 T Bm1 ψ( ), D j 2 E ν2 T Bm2 ψ( ) = C det j 2 F j (ω)e 2πiBmω dω =0.

8 1396 Guochang Wu and Dengfeng Li If j 1 >j 2, then it follows from (3.12) that (3.13) D j 1 E ν1 T Bm1 ψ( ), D j 2 E ν2 T Bm2 ψ( ) = D j 2 E ν2 T Bm2 ψ( ), D j 1 E ν1 T Bm1 ψ( ) =0. Comining with (3.11)-(3.13), we otain that the wave packet system {D j E νt Bm ψ(x)} j Z,,ν S is orthogonal. Therefore, the proof of Theorem 3.1 is completed. Remark 3.1. In particular, let e the elementary matrix E and S =CZ n in Theorem 3.1. Then, we otain Theorem 2.1 in [13], which characterizes the orthogonality of Gaor systems in terms of the Fourier transforms of the functions, that is Corollary 3.1. Let B, C GL n (R). Then, the Gaor system {E Ck T Bm ψ(x)} k, is orthogonal if and only if oth of the equation (3.14) ˆψ(ω + B m) 2 = ψ 2,a.e. ω R n and the equation (3.15) ˆψ(ω + B m Ck) ˆψ(ω + B m)=0,a.e.ω R n hold for every k 0. Remark 3.2. It is immediate to see that if S = {0} in Theorem 3.1, then we otain a sufficient and necessary condition of the orthogonal wavelet system as the following, which is the generalization of the equalities (1.1) and (1.2) in [6, Chapter 3, page 124]. Corollary 3.2. Let e an aritrary matrix, B GL n (R) and ψ L 2 (R n ). Then the wavelet system {D j T Bmψ(x)} j Z, is orthogonal if and only if oth of the equation (3.16) ˆψ(ω + B m) 2 = ψ 2,a.e. ω R n and the equation (3.17) ˆψ( j (ω + B m)) ˆψ(ω + B m)=0,j N, a.e. ω R n hold.

9 Characterizations of the Multivariate Wave Packet Systems 1397 Remark 3.3. systems. Furthermore, we can classify the orthonormality of wave packet Corollary 3.3. Let E n and B GL n (R). Wave packet system {D j E νt Bm ψ(x)} j Z,,ν S is orthonormal if and only if oth of the equality (3.18) ˆψ(ω + B m) 2 =, a.e. ω R n and the equality (3.19) ˆψ( j (ω + B m) ν) ˆψ(ω + B m)=0,j N, a.e. ω R n hold, where = detb and ν 0. Remark 3.4. In the following, we will classify the wave packet systems Ψ 1 and Ψ 2. For wave packet systems Ψ 1, y Lemma 2.4, we get (3.20) D j T BmE ν ψ(x) =e 2πiBm ν D j E νt Bm ψ(x). Then, it deduces from Theorem 3.1 and (3.20) that Corollary 3.4. Let E n and B GL n (R). Wave packet system {D j T BmE ν ψ(x)} j Z,,ν S is orthogonal if and only if oth of the equation (3.21) ˆψ(ω + B m) 2 = ψ 2,a.e. ω R n and the equation (3.22) ˆψ( j (ω + B m) ν) ˆψ(ω + B m)=0,j N, a.e. ω R n hold, where = detb and ν 0. For wave packet systems Ψ 2, it follows from (2) in Lemma 2.4 that (3.23) E ν D p T Bm ψ(x)=d p E νt Bm ψ(x). Thus from theorem 3.1 and (3.23), we have Corollary 3.5. Let E n and B GL n (R). Wave packet system {E ν D j T Bmψ(x)} j Z,,ν S is orthogonal if and only if oth of the equation (3.24) ˆψ(ω + B m) 2 = ψ 2,a.e. ω R n

10 1398 Guochang Wu and Dengfeng Li and the equation (3.25) ˆψ( j (ω + B m) ν) ˆψ(ω + B m)=0,j N, a.e. ω R n hold, where = detb and ν 0. In what follows, we will consider the completeness of wave packet systems in some special cases. In the same way, we use some techniques of wavelet theory. In order to make the prolems more simpler, we pose the condition e 2πiBmCk =1in the theorem. Theorem 3.2. Let E n and B, C GL n (R). Suppose that the function ψ L 2 (R n ) satisfies the equalities (3.26) and j Z k Z n ˆψ( j ω Ck) 2 = a.e.ω R n (3.27) + r=0 k Z n ˆψ( r ω Ck) ˆψ( r (ω + B α) Ck) =0, a.e. ω R n, where = detb and α Z n / Z n. Then the wave packet system {D j E CkT Bm ψ(x)} j Z,k, is complete in L 2 (R n ) when e 2πiBmCk =1. Proof. Let W j,k = span{d j E CkT Bm ψ(x):m Z n },andq j,k denote the orthogonal projection onto the space W j,k. Then, for every f L 2 (R n ),wehave (3.28) Q j,k f(x) = f, D j E CkT Bm ψ D j E CkT Bm ψ(x). In order to prove the completeness of wave packet system {D j E CkT Bm ψ(x)} j Z,k,, it is sufficient to show that for any f L 2 (R n ), (3.29) j Z k Z n (Q j,k f)ˆ(ω) = ˆf(ω). By Lemmas 2.1 and 2.2, it suffices to prove that aove equality holds for f Γ defined y (2.6). For f Γ, we assert (3.30) (Q j,k f)ˆ(ω)= 1 ˆf (ω + j B s) ˆψ( j ω + B s Ck) ˆψ( j ω Ck).

11 Characterizations of the Multivariate Wave Packet Systems 1399 By Plancherel theorem and Lemma 2.4, we otain (3.31) f, D j E CkT Bm ψ = det j 2 ˆf(ω) ˆψ( j ω Ck)e 2πiBm( j ω Ck) dω R n ( ) = det j 2 ˆf( j (ω + B s)) ˆψ(ω + B s Ck) e 2πiBmω dω, where the final equation use e 2πiBmCk = 1. That is to say, the sequences { f, D j E CkT Bm ψ } is evidently the Fourier coefficients of the B ([0, 1] n ) periodic function (3.32) det j 2 ˆf( j (ω + B s)) ˆψ(ω + B s Ck). Thus, we have (3.33) ˆf(ω + j B s) ˆψ( j ω + B s Ck) = ˆf,(D j det j E CkT Bm ψ)ˆ e 2πiBm jω. 2 Multiplying oth sides of (3.33) y ˆψ( j ω Ck) and again using e 2πiBmCk =1, we have ˆf (ω + j B s) ˆψ( j ω + B s Ck) ˆψ( j ω Ck) (3.34) = ˆf,(D j E CkT Bm ψ)ˆ (D j E CkT Bm ψ)ˆ(ω). It deduces from (3.28) and (3.34) that the assertion holds. To clarify the meaning of the series in the aove equations, we point out that the interchange of the order of summation and integration is valid, since we have assumed that function ˆf is compactly supported, consequently, the sums over s, m are finite. By (3.26) and (3.30), we can write the projections Q j,k : (3.35) (Q j,k f)ˆ(ω) j Z k Z n = 1 ˆf(ω) ˆψ( j ω Ck) 2 j Z k Z n + 1 ˆf(ω + j B s) ˆψ( j ω + B s Ck) ˆψ( j ω Ck) j Z k Z n s 0 ( ) = ˆf(ω)+ 1 ˆf(ω+ j B s) ˆψ( j ω+b s Ck)ˆψ( j ω Ck). j Z s 0 k Z n

12 1400 Guochang Wu and Dengfeng Li In the following, we will calculate the second term of the last equality in (3.35). It is not difficult to see that there exists only one non-negative integer r such that s = r α if s =(s 1,s 2,,s n ) (0, 0,, 0), where α Z n / Z n. Set ℵ =: Z n / Z n, then we have (3.36) = = =0. ˆf(ω) ( s 0 j Z ˆf(ω + j B s) R n 1 R n 1 + k Z n ) ˆψ( j ω Ck) ˆψ( j ω + B s Ck) dω ˆf(ω) ( r=0 α ℵ j Z ˆf(ω + j B r α) ) ˆψ( j ω Ck) ˆψ( j ω+b r α Ck) dω k Z n ˆf(ω) ( α ℵ p Z ˆf(ω+ p B α) R n 1 + r=0 ) nˆψ( r p ω Ck) ˆψ( r ( p ω+b α) Ck) dω k Z So, it follows from (2.27) and (3.36) that (3.29) holds. The proof of Theorem 3.2 is finished. 4. THE NECESSRY CONDITION ND SUFFICIENT CONDITION OF THE WVE PCKET PRSEVL FRME In this section, we will give the necessary condition and sufficient condition for the wave packet system to e a Parseval frame with the very general lattices. Then, we otain the corresponding known results in Gaor system and wavelet system as some corollaries. First, we estalish a lemma as follows. Lemma 4.1. Suppose that wave packet system {D p E ν T Bm ψ(x)}, is defined y (2.3), then for any f Γ, (4.1) = f, D p E ν T Bm ψ 2 m Z n q p ˆf ( p (ω + B s + ν)) ˆψ(ω + B s) 2 dω, where = detb and q p = det p.

13 Characterizations of the Multivariate Wave Packet Systems 1401 Proof. Let f Γ, then ˆf C c (R) and ˆf have compact support. Put q p = det p, then y Lemma 2.3 and Plancherel theorem, we have f, D p E ν T Bm ψ 2 = (4.2) ˆf, 2 D T p ν E Bm ˆψ = q p ˆf( p (ω + ν)) ˆψ(ω)e 2πiBmω dω 2. p P ν Q R n We affirm: q p ˆf( p (ω + ν)) ˆψ(ω)e 2πiBmω dω 2 p P ν Q m Z (4.3) n R n = q p ˆf ( p (ω + B s + ν)) ˆψ(ω + B s) 2 dω. Now, we prove (4.3). Fix (p, ν) S. The interchange of the order of summation and integration is valid, since we have assumed that function ˆf is compactly supported, and, consequently the sum over m is finite. Similar to (3.6)-(3.9), we can define the function F p : R C y (4.4) F p (ω) = ˆf( p (ω + B s + ν)) ˆψ(ω + B s), a.e.ω and prove that F p (ω) L 1 (B [0, 1] n ). lso, we can otain F p (ω) L 2 (B [0, 1] n ). Then, according to the definition of F p (ω), wehave ˆf( p (ω + ν)) ˆψ(ω)e 2πiBmω dω R n ( ) (4.5) = ˆf( p (ω + B s + ν)) ˆψ(ω + B s) e 2πiBmω dω = F p (ω)e 2πiBmω dω. Parseval s equality shows that (4.6) F p (ω)e 2πiBmω dω 2 = 1 F p (ω) 2 dω. Comining (4.5),(4.6) and the definition of F p (ω), weget ˆf( p (ω + ν)) ˆψ(ω)e 2πiBmω dω 2 m Z (4.7) n R n = 1 ˆf( p (ω + B s + ν)) ˆψ(ω + B s) 2 dω.

14 1402 Guochang Wu and Dengfeng Li So, (4.3) holds. t last, comparing to (4.2) and (4.3), we find that (4.1) holds. Based on Lemma 4.1, we will estalish a sufficient condition for the wave packet system to e a Parseval frame. Theorem 4.1. The wave packet system {D p E ν T Bm ψ(x)}, defined y (2.3) is a wave packet Parseval frame for L 2 (R n ) if the two following equalities (4.8) ˆψ( pω ν) 2 =, a.e. ω R n and (4.9) ν Q ˆψ( pω ν) ˆψ( pω + B s ν) =0, hold for every s Z n and s 0,where = detb. a.e. ω R n Proof. In the same way, let f Γ. Notice that function ˆf in the series f, D p E ν T Bm ψ 2 is compactly supported, hence the sum over m is finite, consequently, the interchange of the order of summation and integration is valid. By Lemma 2.4, Lemma 4.1 and Plancherel theorem, we can write f, D p E ν T Bm ψ 2 = q p ˆf( p (ω + B s + ν)) ˆψ(ω + B s) 2 dω = q p ( ˆf( p (ω + B s + ν)) ˆψ(ω + B s)) ( ˆf( (4.10) p (ω + B m + ν)) ˆψ(ω + B m))dω = q p ˆf( p (ω + ν) ˆψ(ω)( nˆf( p (ω + B s+ν)) ˆψ(ω+B s)dω R n s Z = q p ˆf( p ω) ˆψ(ω ν) 2 dω R n + q p ˆf( p ω) ˆψ(ω ν)( ˆf( p(ω+b s) ˆψ(ω+B s ν))dω. R n s 0 For future reference, we introduce the following notations: (4.11) I(f) = f, D p E ν T Bm ψ 2,

15 Characterizations of the Multivariate Wave Packet Systems 1403 (4.12) I 1 (f) = and q p R n ˆf( p(ω)) ˆψ(ω ν) 2 dω (4.13) I 2 (f) = q p R n ˆf( p ω) ˆψ(ω ν)( s 0 ˆf( p(ω+b s) ˆψ(ω+B s ν))dω. Oviously, from (4.11)-(4.13), we have the following decomposition (4.14) I(f) =I 1 (f)+i 2 (f). By f Γ and the inequality (4.15) ˆψ(ω ν) ˆψ(ω + B s ν)) 1 2 ( ˆψ(ω ν) 2 + ˆψ(ω + B s ν)) 2 ), we have (4.16) R n ˆf ( p (ω)) ˆψ(ω ν)( s 0 ˆf( p(ω+b s)) ˆψ(ω+B s ν))dω <. Thus, we can change the orders of integration and summation in the expression (4.13). Using the equations (4.8) (4.9) and (4.10), we get f, D p E ν T Bm ψ 2 m Z n q p = R ˆf( pω) 2 ( ˆψ(ω ν) 2 )dω n p P ν Q q p + R ˆf( pω) ˆf( p(ω+b s))( ˆψ(ω ν) ˆψ(ω+B s ν))dω (4.17) n p P s 0 ν Q 1 = R ˆf(ω) 2 ( ˆψ( pω ν) 2 )dω n p P ν Q 1 + R ˆf(ω)( ˆf(ω+ pb s) ˆψ( pω ν) ˆψ( pω+b s ν))dω n s 0 p P ν Q = ˆf(ω) 2 dω = f 2. R n So, it follows from Lemmas 2.1 and 2.2 that the wave packet system {D p E ν T Bm ψ(x)}, defined y (2.3) is a Parseval frame. The proof of Theorem 4.1 is completed. Remark 4.1. Let e the elementary matrix E in Theorem 4.1. Then we otain a sufficient condition of the Gaor Parseval frame as follows.

16 1404 Guochang Wu and Dengfeng Li Corollary 4.1. Let B, C GL n (R). Then, the Gaor system {E Ck T Bm ψ(x)} k, is a Parseval frame for L 2 (R n ) if the two following equalities ˆψ(ω Ck) 2 =, a.e. ω R n k Z n and k Z n ˆψ(ω Ck) ˆψ(ω + B s Ck) =0, a.e. ω R n hold for every s Z n and s 0,where = detc. lso, if P = { j : j Z, GL n (R)} and Q = {0} in Theorem 4.1, then we otain the sufficient of the wavelet frame, that is Corollary 4.2. Let e an aritrary matrix and ψ L 2 (R n ). Then the wavelet system {D j T Bmψ(x)} j Z, is a Parseval wavelet frame if the two following equalities ˆψ( j ω) 2 =, a.e. ω R n and j Z ˆψ( ω) ˆψ( j ω + B s)=0, hold for every s Z n and s 0,where = detb. j N, a.e. ω R n pplying some techniques in [6, 7], we can also provide a necessary condition for the wave packet system to e a Parseval frame in L 2 (R n ). Theorem 4.2. If the wave packet system {D p E ν T Bm ψ(x)}, defined y (2.3) is a Parseval frame for L 2 (R n ), then the equalities (4.8) and (4.18) (p,ν) S ˆψ( pω ν) ˆψ( pω + B s ν) =0, hold for every s Z n and s 0,where = detb. a.e. ω R n (4.19) Proof. Now we assume that f, D p E ν T Bm ψ 2 = f 2 for all f L 2 (R n ). In particular, (4.19) holds for all functions f Γ. It is ovious that for f Γ, I(f) < if and only if I 1 (f) <. Now taking f = χ C,whereC is any compact suset in R n, we see that I 1 (f) < for all f Γ,

17 if and only if τ(ω) := Characterizations of the Multivariate Wave Packet Systems 1405 ˆψ( p ω ν) 2 is locally integrale in R n. Thus, y our assumption, the function τ(ω) is locally integrale, consequently, almost every point of the function τ is a Leesgue point. Choose ω 0 R to e a Leesgue point of the function τ(ω). Thenwehave (4.20) ˆψ( pω 0 ν) 2 = lim ɛ 0 ω ω 0 ɛ 1 B(ɛ) ˆψ( pω ν) 2 dω, where B(ɛ) denotes the all of radius ɛ>0 aout the origin. For small enough ɛ, we define f ɛ y (4.21) ˆfɛ (ω) = Clearly, 1 B(ɛ) χ B(ɛ) (ω ω 0 ). (4.22) f ɛ 2 = ˆf ɛ 2 =1. By the decomposition (4.14), we have (4.23) I(f ɛ )=I 1 (f ɛ )+I 2 (f ɛ ). Thus, y using (4.20)-(4.23) and the definition of I 1,wefind (4.24) 1 = 1 τ(ω ω 0 )dω + I 2 (f ɛ ). B(ɛ) B(ɛ) If we can show that lim I 2 (f ɛ )=0,thensinceω 0 is a Leesgue point of τ, we ɛ 0 have 1 (4.25) 1 = lim( τ(ω ω 0 )dω + I 2 (f ɛ )) = τ(ω 0 ) + lim I 2 (f ɛ )=τ(ω 0 ). ɛ 0 B(ɛ) B(ɛ) ɛ 0 That is to say, τ(ω) =1, a.e. ω R n, which completes the proof of (4.8). Now, we devote to calculating lim I 2 (f ɛ ). ɛ 0 (4.26) I 2(f ɛ) q p ˆf ɛ( p(ω)) ˆψ(ω ν) ˆf ɛ( p(ω + B s) ˆψ(ω + B s ν)) dω R n s 0 1 = R n ˆf ɛ(ω) ˆf ɛ(ω + pb s) ( ˆψ( pω ν) ˆψ( pω + B s ν)) dω p P s 0 ν Q 1 R n ˆf ɛ(ω) ˆf ɛ(ω+ pb s) ( ( ˆψ( pω ν) 2 + ˆψ( pω+b s ν)) 2 )dω p P s 0 ν Q 1 = R n ˆf ɛ(ω) ˆf ɛ(ω + pb s) ˆψ( pω ν) 2 dω p P s 0 ν Q 1 + ˆf ɛ(ω) ˆf ɛ(ω + pb s) ˆψ( pω + B s ν)) 2 dω. R n p P s 0 ν Q

18 1406 Guochang Wu and Dengfeng Li Since ɛ is small enough, for every s 0,wehave (4.27) ˆf ɛ (ω) ˆf ɛ (ω + pb s) =0. Thus, for small enough ɛ, I 2 (f ɛ )=0. In the following, we turn to prove (4.18). ccording to (4.14) and (4.19), for all f Γ, we easily deduce I 2 (f) =0, i.e. (4.28) R n p P 1 ˆf(ω) ˆf(ω+ pb s)( ˆψ( pω ν) ˆψ( pω + B s ν))dω =0. s 0 ν Q By the polarization identity, for any f, g Γ, weget 1 (4.29) ˆf(ω)ĝ(ω+ p B s)( ˆψ( p ω ν) ˆψ( pω+b s ν))dω =0. R n p P s 0 ν Q Similar to discussion of aove function τ(ω), the function (4.30) μ(ω) := ˆψ( pω ν) ˆψ( pω + B s ν) ν Q is locally integrale. Hence, almost every point of the function ν(ω) is a leesgue point in R n. Fix s 0 0. Let ω 0 (ω 0 0and ω 0 + s 0 0) e a Leesgue point of μ(ω), and f ɛ e defined y (4.21) and h e defined as the following (4.31) ĥ ɛ (ω) = Then, we have 1 B(ɛ) χ B(ɛ) (ω ω 0 pb s 0 ). (4.32) ˆfɛ (ω ω 0 )ĥɛ(ω ω 0 + pb s 0 )= 1 B(ɛ) χ B(ɛ) (ω ω 0 ). Let f = f ɛ and g = h ɛ in (4.29), then we get 1 0= R ˆf ɛ (ω)ĝ ɛ (ω+ pb s)( ˆψ( pω ν) ˆψ( pω+b s ν))dω n p P s 0 ν Q 1 (4.33) = ˆψ( pω ν ω 0 ) ˆψ( pω + B s 0 ν ω 0 )dω B(ɛ) B(ɛ) p P ν Q 1 + ˆf ɛ (ω)ĝ ɛ (ω+ pb s)( ˆψ( pω ν) ˆψ( pω +B s ν))dω. R n p P s 0,s 0 ν Q It is ovious that for small enough ɛ, 1 (4.34) ˆf ɛ (ω)ĝ ɛ (ω+ pb s)( ˆψ( pω ν) ˆψ( pω+b s ν))dω =0. R n p P s 0,s 0 ν Q

19 Characterizations of the Multivariate Wave Packet Systems 1407 Therefore, y the fact that ω 0 is a Leesgue point of μ(ω), 1 0 = lim ɛ 0 ˆψ( pω ν ω 0 ) ˆψ( pω + B s 0 ν ω 0 )dω B(ɛ) B(ɛ) p P ν Q (4.35) = 1 ˆψ( pω ν) ˆψ( pω + B s 0 ν)dω, p P ν Q for all s 0 and a.e. ω R n. So, we get (4.18). The proof of Theorem 4.2 is ended. Remark 4.2. If is the elementary matrix E in Theorem 4.2, then, we otain the necessary condition of the Gaor frames as follows. Corollary 4.3. Let B, C GL n (R). If the Gaor system {E Ck T Bm ψ(x)} k, is a Gaor Parseval frame for L 2 (R n ), then the two following equalities ˆψ(ω Ck) 2 =, a.e. ω R n k Z n and k Z n ˆψ(ω Ck) ˆψ(ω + B s Ck) =0, a.e. ω R n hold for every s Z n and s 0,where = detb. lso, if P = { j : j Z, GL n (R)} and Q = {0} in Theorem 4.2, then we otain the following necessary condition of the wavelet frames. Corollary 4.4. Let e an aritrary matrix, B GL n (R) and ψ L 2 (R n ). If the wavelet system {D j T Bmψ(x)} j Z, is a Parseval wavelet frame, then the two following equalities ˆψ( j ω) 2 =, a.e. ω R n and j Z ˆψ( ω) ˆψ( j ω + B s)=0, j Z hold for every s Z n and s 0,where = detb. a.e. ω R n Remark 4.3. Comparing with corollary 4.1 and corollary 4.3, we otain the following characterization of the Gaor Parseval frame, which is the case of single generator of theorem 3.1 in [13]. Corollary 4.5. Let B, C GL n (R). Then, the Gaor system {E Ck T Bm ψ(x)} k, is a Gaor Parseval frame for L 2 (R n ) if and only if the two following equalities ˆψ(ω Ck) 2 =, a.e. ω R n k Z n

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21 Characterizations of the Multivariate Wave Packet Systems E. Hernández and G. Weiss. First Course on Wavelets, CRC Press, Boca Raton, FL, Guochang Wu Department of Mathematics Henan University of Economics and Law Zhengzhou P. R. China Dengfeng Li School of Mathematics and Computer Science Wuhan Textile University Wuhan P. R. China