SUMS OF MATRIX-VALUED WAVE PACKET FRAMES IN

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1 GLASNIK MATEMATIČKI Vol. 53(73)(018), SUMS OF MATRIX-VALUED WAVE PACKET FRAMES IN L (R d,c s r ) Jyoti, Deepshikha, Lalit K. Vashisht and Geetika Verma University of Delhi, India and University of South Australia, Australia Abstract. The purpose of this paper is to first show relations between wave packet frame bounds and the scalars associated with finite sum of matrix-valued wave packet frames for the matrix-valued function space L (R d,c s r ). A sufficient condition with explicit wave packet frame bounds for finite sum of matrix-valued wave packet frames in terms of scalars and frame bounds associated with the finite sum of frames is given. An optimal estimate of wave packet frame bounds for the finite sum of matrix-valued wave packet frames is presented. In the second part, we show that the rate of convergence of the frame algorithm can be increased by using frame bounds and scalars associated with the finite sum of frames. Finally, a necessary and sufficient condition for finite sum of matrix-valued wave packet frames in terms of series associated with wave packet vectors is given. 1. Introduction Cordoba and Fefferman in [6] introduced the concept of wave packet system by applying certain collection of dilations, modulations and translations to the Gaussian function in the study of some classes of singular integral operators. In mathematical physics, a wave packet is a function Ψ defined on a domain Ω R d which is well localized in phase space. That is, Ψ and its Fourier transform Ψ are both concentrated in reasonably small sets. Wave packets over the real line are generated from classical dilations, translations 010 Mathematics Subject Classification. 4C15, 4C30, 4C40. Key words and phrases. Bessel sequence, frames, wave packet frames. The research of Jyoti is supported by Council of Scientific and Industrial Research (CSIR), India (Grant No.: 09/045(1374)/015-EMR-I). Deepshikha is supported by CSIR, India (Grant No.: 09/045(135)/014-EMR-I). Lalit was partially supported by R & D Doctoral Research Programme, University of Delhi (Grant No.: RC/014/600). 153

2 154 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA and modulations of a given window (or scaling) function. Labate et al. in [3] adopted the same expression to describe, more generally, any collection of functions which are obtained by applying the same operations to a finite family of functions in L (R d ). Gabor systems, wavelet systems and the Fourier transform of wavelet systems are special cases of wave packet systems. Lacey and Thiele ([4, 5]) gave applications of wave packet systems in boundedness of the Hilbert transforms. The wave packet systems and related frame properties have been studied by several authors, see [5,7,10,13,18 0] and references therein. Recently, Obeidat, Samarah, Casazza and Tremain studied finite sum of Hilbert space frames in[6]. They discussed finite sum of frames which include images of Bessel sequences and frames under bounded linear operators and frame operator on the underlying space. In the present work, we consider a matrix-valued Weyl-Heisenberg wave packet frame of the form W(Ψ) {D Aj T Bk E Cm Ψ} forthe matrix-valuedfunction spacel (R d,c s r ), seedefinition 3.1. Notable contribution in the paper includes necessary and sufficient conditions with explicit frame bounds for the finite sum of matrix-valued wave packet frames for the function space L (R d,c s r ) in terms of wave packet frame bounds and scalars associated with the given finite sum of matrix-valued wave packet frames. It is shown that the width of the frame can be decreased by using frame bounds and scalars associated with the finite sum of frames. This is useful in the frame algorithms which depend on the frame bounds. Optimal frame bounds of matrix-valued wave packet frames are discussed Frames in Hilbert Spaces. Let H be a separable complex Hilbert space with an inner product.,. and norm. =.,.. A countable sequence {f k } k I H is a frame for H if there exist positive real numbers L o U o < such that (1.1) L o f k I f,f k U o f for all f H. The scalars L o and U o are called lower frame bound and upper frame bound, respectively. If it is possible to choose L o = U o, then we say that the frame is tight. It is Parseval if L o = U o = 1. The sequence {f k } k I is called a Bessel sequence with Bessel bound U o if the upper inequality in (1.1) holds for all f H. The ratio U o L o U o +L o is called the width of the frame. We recall that the width of a frame measures its tightness.

3 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 155 The scalars L Opt = sup { } L o > 0 : L o satisfies lower inequality in (1.1) and { } U Opt = inf U o > 0 : U o satisfies upper inequality in (1.1) are called the best frame bounds or optimal frame bounds of the frame. Let {f k } k I be a frame for H. The following three operators play an important role in stable reconstruction and analysis of each vector (signal) in the underlying space: T : l (I) H, T{c k } k I = k I c k f k, {c k } k I l (I) (pre-frame operator), T : H l (I), T f = {f,f k } k I, f H (analysis operator), S = TT : H H, Sf = k If,f k f k, f H (frame operator). The frame operator S is a bounded, linear, invertible and positive operator on H. This gives the reconstruction of each vector f H, f = SS 1 f = k I S 1 f,f k f k = k If,S 1 f k f k. Thus, a frame for H allows each vector in H to be written as a linear combination of the elements in the frame, but the linear independence between the frame elements is not required. More precisely, frames are basis-like building blocks that span a vector space but allow for linear dependency, which is useful to reduce noise, find sparse representations, spherical codes, compressed sensing, signal processing, wavelet analysis etc., see [1]. For recent development in discrete frames in the unitary space C N, we refer to [11,1]. Nowadays frames are also studied by using the iterated function systems (IFS) ([15, 7, 8]), reproducing systems ([16]), quantum systems ([9]). Duffin and Schaeffer in [14] described the frame condition for the first time, in the context of non-harmonic Fourier series, as a method to calculate the coefficients in a linear combination of the vectors of a linearly dependent spanning set, a Hilbert space frame. Few years later, in 1986, frames were reintroduced and brought to life by Daubechies, Grossmann and Meyer ([8]). Since then, the theory of frames began to be studied more widely. The basic theory of frames and their applications in different directions in science and engineering can be found in the books of Casazza and Kutyniok ([1]), Christensen ([4]), Heil ([17]), Daubechies ([9]) and beautiful research tutorials by Casazza ([]) and Casazza and Lynch ([3]).

4 156 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA 1.. Related work and Motivation. Holub proved in [1] that if {e k } k=1 is any normalized basis for a separable Hilbert space H and {fk } k=1 is the associateddual basis of coefficient functionals, then the sequence {f k +fk } k=1 is again a basis for H. Obeidat, Samarah, Casazza and Tremain studied finite sum of Hilbert space frames in [6]. They proved necessary and sufficient conditionsonbesselsequences{f k } k=1 and{g k} k=1 andboundedlinearoperators L 1, L on a separable Hilbert space H such that the sum {L 1 f k +L g k } k=1 is a frame for H. Kaushik, Singh and Virender gave sufficient conditions for finite sum of Gabor frames for L (R) in terms of frame bounds and scalars associated with the finite sum of Gabor frames, see Theorem 4.3 of []. It will be interesting to find new relations and conditions on scalars and frame bounds associated with the finite sum of frames for the underlying space. The frame algorithm (see Theorem 4.1) gives approximation of each signal (vector) in the space. Furthermore, the convergence rate of the frame algorithm depends on the width of the frame, i.e., on the frame bounds. Naturally, it is an important consideration that to what extent the width of frame can be decreased and range of the optimal frame bounds can be minimized. This is our motivation for writing this paper Outline. The rest of the paper is organized as follows: In Section, we recapitulate basic facts about matrix-valued wave packet frames in the matrix-valued function space L (R d,c s r ). Section 3 starts with a sufficient condition with explicit frame bounds for finite sum of matrix-valued wave packet frames for L (R d,c s r ), see Theorem 3.. It is observed that the condition given in Theorem 3. is only sufficient but not necessary, see Example 3.3. A necessary and sufficient condition for finite sum of matrix-valued wave packet frames in terms of series associated with wave packet vectors is given in Theorem 3.4. An example spinning off of Theorem 3.4 is given. An optimal estimate of wave packet frame bounds for the finite sum of matrix-valued wave packet frames is presented in Theorem 3.9. In Section 4, we discuss an application of frame bounds associated with sums of matrix-valued wave packet frames in separable Hilbert spaces. It is shown that the rate of approximation in the frame algorithm can be increased by suitable choice of scalars and frame bounds which appear in the finite linear sum of frames, see Example 4... Preliminaries Throughout the paper, symbols Z, N, R and C denote the set of integers, positive integers, real numbers and complex numbers, respectively. Let d,s,r N. The complex vector space of all s-by-s complex matrices is denoted by M s (C). Matrix-valued functions are denoted by bold letters. We

5 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 157 consider the matrix-valued function space { } L (R d,c s r ) = f(u) : u R d, f ij (u) L (R d ) (1 i s,1 j r), where f 11 (u) f 1 (u) f 1r (u) f 1 (u) f (u) f r (u) f(u) = f s1 (u) f s (u) f sr (u) The norm on L (R d,c s r ) is given by ( f = f ij (u) du R d 1 i s 1 j r The integration of a matrix-valued function f L (R d,c s r ) is given by R f d 11 (u) f(u) = R f d 1 (u). R d. f R d s1 (u) )1 R f d 1 (u) R f d 1r (u) R f d (u) R f d r (u) f R d s (u) f R d sr (u) The matrix-valued inner product on L (R d,c s r ) is defined as f,g = f(u)g (u), f,g L (R d,c s r ), R d where denotes the transpose and the complex conjugate. It has the following properties. 1. For every f,g L (R d,c s r ), f,g = g,f.. For every f,g,h L (R d,c s r ) and for every M 1,M M s (C), It is also readily seen that M 1 f+m g,h = M 1 f,h +M g,h. f = tracef,f. We consider the space l (Z d+,m s (C)) given by { } l (Z d+,m s (C)) := {M k } k Z d+ M s (C) : M k <. k Z d+.

6 158 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA l (Z d+,m s (C)) is a Banach space with respect to the norm given by ( )1 {M k } k Z d+ =. k Z d+ M k By GL d (R) we denote the set of all invertible d-by-d matrices over R. Let a,b R d and let C be a real d-by-d matrix. For b and u in R d, b u denotes the standard inner product of b and u. We consider operators T a, E b,d C : L (R d,c s r ) L (R d,c s r ) given by T a f(u) = f(u a) (Translation by a), E b f(u) = e πib u f(u) (Modulation by b), D C f(u) = detc 1 f(cu) (Dilation by C). We end the section by recording two lemmas which give Cauchy-Schwartz type inequality and norm of an operator on L (R d,c s r ), respectively. Lemma.1. For any f,g L (R d,c s r ), we have tracef,g f g. Lemma.. Let U : L (R d,c s r ) L (R d,c s r ) be a bounded linear operator such that Then Uf,g = f,ug, for all f,g L (R d,c s r ). U = sup traceuf,f. f =1 f L (R d,c s r ) 3. Main Results Let Ψ L (R d,c s r ) and let Λ 1,Λ R and Λ d R d be countable sets. Let {A j } j Λ1 GL d (R), B GL d (R) and {C m } m Λ R d. A system of the form W(Ψ) {D Aj T Bk E Cm Ψ} j Λ1,m Λ,k Λ d iscalledaweyl-heisenberg matrix-valued wave packet system inl (R d,c s r ). In thispaper, weconsiderλ 1 = Λ = Z,Λ d = Z d sothat the familyconsidered is W(Ψ) = {D Aj T Bk E Cm Ψ}. Definition 3.1. A family W(Ψ) {D Aj T Bk E Cm Ψ} in L (R d,c s r ) is a Weyl-Heisenbergmatrix-valuedwavepacketframe (WHMV wave packet frame, in short) for L (R d,c s r ) if there exist positive scalars L Ψ,U Ψ < such that for all f L (R d,c s r ), L Ψ f (3.1) f,d Aj T Bk E Cm Ψ UΨ f.

7 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 159 The scalars L Ψ and U Ψ are called lower wave packet frame bound and upper wave packet frame bound, respectively. If the upper inequality in (3.1) is satisfied,thenwesaythatw(ψ)isawhmv wave packet Bessel sequence with Bessel bound U Ψ. The frame operator S Ψ : L (R d,c s r ) L (R d,c s r ) associated with W(Ψ) is given by S Ψ : f f,d Aj T Bk E Cm Ψ D Aj T Bk E Cm Ψ, f L (R d,c s r ). If W(Ψ) is a frame for L (R d,c s r ), then S Ψ is a linear, bounded and invertible operator satisfying and hence S Ψ f,g = f,s Ψ g, f,g L (R d,c s r ) S 1 Ψ f,g = f,s 1 Ψ g, f,g L (R d,c s r ). Let [p] = {1,,3,..,p} be a finite subset of N and let {A j } j Z GL d (R), B GL d (R), {C m } m Z R d. For each ξ [p], assume that W(Ψ ξ ) {D Aj T Bk E Cm Ψ ξ } is a WHMV wavepacket frame for L (R d,c s r ), where Ψ ξ L (R d,c s r ). The finite sum { p } α ξ W(Ψ ξ ) p α ξ D Aj T Bk E Cm Ψ ξ, where α 1,α,...,α p are non-zero scalars, in general, does not constitute a frame for L (R{ d,c s r ). The following theorem gives a sufficient condition for p } the finite sum α ξw(ψ ξ ) in terms of wave packet frame bounds and scalars associated with the finite sum to be a WHMV wave packet frame for the function space L (R d,c s r ). An application of this result to the frame algorithm is given in Section 4. In matrix theory this type of condition is known as the dominance property. Theorem 3.. For each ξ [p], let W(Ψ ξ ) = {D Aj T Bk E Cm Ψ ξ } be a WHMV wave packet frame for L (R d,c s r ) with frame bounds L ξ,u ξ and let α 1,α,...,α p be non-zero scalars. If (3.) α i L i > p α ξ U ξ + ξ i p ξ i,t i,ξ t α ξ α t U ξ U t

8 160 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA for some i [p], then the finite sum L (R d,c s r ) with frame bounds α i p L i α ξ U ξ + ξ i { p } α ξw(ψ ξ ) p ξ i,t i,ξ t is a frame for α ξ α t U ξ U t, and α i p U i + α ξ U ξ + ξ i p ξ i,t i,ξ t α ξ α t U ξ U t. Proof. By using Cauchy-Bunyakovsky-Schwarz inequality, we compute j,m Z k Z d f, = j,m Z k Z d + p α ξ D Aj T Bk E Cm Ψ ξ α i D Aj T Bk E Cm Ψ i [ p α ξ f,d Aj T Bk E Cm Ψ ξ ξ i p ξ i,t i,ξ t p α ξ ξ i ] α ξ α t f,d Aj T Bk E Cm Ψ ξ f,d Aj T Bk E Cm Ψ t f,d Aj T Bk E Cm Ψ ξ p + α ξ α t f,d AjT BkE CmΨ ξ f,d AjT BkE CmΨ t j,m Z ξ i,t i,ξ t k Z d p p α ξ U ξ + α ξ α t U ξ U t f, ξ i ξ i,t i,ξ t for any f L (R d,c s r ).

9 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 161 By using the obtained estimate, for any f L (R d,c s r ), we compute 1/ p f, α ξ D Aj T Bk E Cm Ψ ξ j,m Z k Z d f,α i D T Aj Bk E Ψ Cm i j,m Z k Z d + f, j,m Z k Z d 1/ p α ξ D Aj T Bk E Cm Ψ ξ α i D Aj T Bk E Cm Ψ i α i p U i f + α ξ U ξ + ξ i = α i p U i + α ξ U ξ + ξ i This gives, for any f L (R d,c s r ), that p f, α ξ W(Ψ ξ ) (3.3) j,m Z k Z d α i p U i + α ξ U ξ + ξ i p ξ i,t i,ξ t p ξ i,t i,ξ t Next we compute (for any f L (R d,c s r )) p f, α ξ D Aj T Bk E Cm Ψ ξ f,α i D T Aj Bk E Ψ Cm i f, p ξ i,t i,ξ t 1/ 1/ α ξ α t U ξ U t f α ξ α t U ξ U t f. 1/ α ξ α t U ξ U t f. p α ξ D Aj T Bk E Cm Ψ ξ α i D Aj T Bk E Cm Ψ i 1/

10 16 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA α i p L i f α ξ U ξ + ξ i = α i p L i α ξ U ξ + ξ i p ξ i,t i,ξ t p ξ i,t i,ξ t Therefore, for all f L (R d,c s r ), we have (3.4) f, p α ξ W(Ψ ξ ) α i p L i α ξ U ξ + ξ i α ξ α t U ξ U t f α ξ α t U ξ U t f. p ξ i,t i,ξ t { p α ξ α t U ξ U t f. } Hence, by (3.3) and (3.4), the finite sum α ξw(ψ ξ ) is a WHMV wave packet frame for L (R d,c s r ) with the required frame bounds. The following example shows that condition (3.) given in Theorem 3. is only sufficient but not necessary. Example 3.3. Let p = and let τ 1 be the triangle of vertices (0,0),(1,0) and (0,1). Consider a sequence of non-degenerate triangles {τ j } j N with disjoint interiors satisfying τ j = R. j N Let u j = (u (1) j,u () j ), v j = (v (1) j,v () j ) and w j = (w (1) j,w () j ) denote the vertices of the triangle τ j for each j N. Then the wave packet system {D Aj E Cj T k ψ} j N,k Z with ˆψ(ξ) = χ τ1 (ξ) for ξ R,C j = (A T j ) 1 u j is a Parseval frame for L (R ), where (see [3]) [ ] A T vj j = (1) (1) u j w (1) (1) j u j v () () j u j w () (). j u j Since D Aj T k E Cj ψ = exp( πik.c j )D Aj E Cj T k ψ, we get that the family {D Aj T k E Cj ψ} j N,k Z is a Parseval frame for L (R ). Let [ ] 0 ψ1 Ψ =, where ψ ψ = ψ 1 = ψ.

11 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 163 Then, Ψ L (R,C ) and for all f = j N,k Z D Aj T k E Cj Ψ,f [ ] f11 f 1 L f 1 f (R,C ), we have = [ D Aj T k E Cj ψf 1 + D Aj T k E Cj ψf j N,k Z R R + D Aj T k E Cj ψf 11 + ] D Aj T k E Cj ψf 1 R R = f 1 + f + f 11 + f 1 = f. Hence, {D Aj T k E Cj Ψ} j N,k Z is a Parseval WHMV wave packet frame for L (R,C ). Choose α 1 = α = 1 and Ψ 1 = Ψ = Ψ. Then, the family W(Ψ ξ ) = {D Aj T k E Cj Ψ ξ } j N,k Z is a WHMV wave packet frame for L (R,C { ) with frame bounds L ξ = 1, } U ξ = 1 (ξ {1,}). Hence the finite sum α ξw(ψ ξ ) is a WHMV wave packet frame for L (R,C ) with frame bounds L o = U o = 4. The estimate in inequality (3.) for i = 1 is given by 1 = α 1 L 1 > α ξ U ξ + ξ 1 ξ 1,t 1,ξ t α ξ α t U ξ U t = α U = 1, a contradiction. Similarly, for i =, we arrive at a contradiction. Kaushik, Singh and Virender in [] studied finite sum of Gabor frames in L (R). They observedthat ifsomescalarmultiple ofaseriesassociatedwith a GaborframeisdominatedbytheseriesassociatedwiththefinitesumofGabor frames, then the finite sum constitutes a Gabor frame for the underlying space and vice-versa, see Theorem 4. of[]. The following theorem generalizes this result in the context of WHMV wave packet frames. This is an adaption of [, Theorem 4.]. Theorem 3.4. For each ξ [p], let W(Ψ ξ ) = {D Aj T Bk E Cm Ψ ξ } be a WHMV wave packet frame for L (R d,c s r ) with frame bounds L ξ,u ξ, and let α 1,α,...,α p be any non-zero scalars. Then, the finite sum { p } α ξ W(Ψ ξ ) is a frame for L (R d,c s r ) if and only if there exists β > 0

12 164 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA and i [p] such that β f,d T Aj Bk E Ψ (3.5) Cm i f, j,m Z k Z d for all f L (R d,c s r ). j,m Z k Z d p α ξ W(Ψ ξ ), Proof. Suppose (3.5) holds. Then, for any f L (R d,c s r ), we have βl i f β f, D Aj T Bk E Cm Ψ i = f, f, p α ξ W(Ψ ξ ) p α ξ D Aj T Bk E Cm Ψ ξ p α ξ U ξ p f. { p } Hence the finite sum α ξ W(Ψ ξ ) is a WHMV wave packet frame for ( L (R d,c s r p ) with frame bounds βl i and α ξ U ξ )p. { p Conversely, let } α ξ W(Ψ ξ ) be a frame for L (R d,c s r ) with frame bounds L o,u o. Then for any (but fixed) i [p], we have 1 f, D Aj T Bk E Cm Ψ i f, f L (R d,c s r ). U i Therefore, for β = Lo U i > 0, we have β f, D Aj T Bk E Cm Ψ i = L o U i L o f f, f, D T Aj Bk E Ψ Cm i p α ξ W(Ψ ξ ) for all f L (R d,c s r ). The proof is complete.

13 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 165 The following example illustrates Theorem 3.4. Example 3.5. Consider the Parseval frame {D Aj T k E Cj ψ} j N,k Z for L (R ) given in Example 3.3. (I): Let Ψ 1 = ψ 0 L (R,C 3 1 ). 0 Then, for all f = f 11 L (R,C 3 1 ), we have j N,k Z f 1 f 31 D Aj T k E Cj Ψ 1,f = [ D Aj T k E Cj ψf 11 + D Aj T k E Cj ψf 1 j N,k Z R R + ] D Aj T k E Cj ψf 31 R = f 11 + f 1 + f 31 = f. Hence, {D Aj T k E Cj Ψ 1 } j N,k Z is a Parseval WHMV wave packet frame for L (R,C 3 1 ). Similarly, we can show that {D Aj T k E Cj Ψ } j N,k Z is a Parseval WHMV wave packet frame for L (R,C 3 1 ), where Ψ = 0. ψ 0 { Choose α 1 = 1 and α = 1 3. Then, } α ξ W(Ψ ξ ) is equal to 1 {D Aj T k E Cj Ψ} j N,k Z,whereΨ = ψ 0. Foranyf = f 11 L (R,C 3 1 ), we compute α ξ W(Ψ ξ ),f j N,k Z = = 1 4 j N,k Z j N,k Z 1 3 ψ D Aj T k E Cj Ψ,f f 1 f 31 [ D Aj T k E Cj ψf 11 + D Aj T k E Cj ψf 1 R R

14 166 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA + ] D Aj T k E Cj ψf [ D Aj T k E Cj ψf 11 R 9 j N,k Z R + D Aj T k E Cj ψf 1 ] + D Aj T k E Cj ψf 31 R R = 1 [ f 11 + f 1 + f 31 ] + 1 [ f 11 + f 1 + f 31 ] 4 9 = f. Choose β = 1 36 and i = 1. Then, for all f L (R,C 3 1 ), we have β j N,k Z 1 D Aj T k E Cj Ψ i,f = 36 f f = j N,k Z { Hence, by Theorem 3.4, the finite sum packet frame for L (R,C 3 1 ). α ξ W(Ψ ξ ),f. } α ξ W(Ψ ξ ) is a WHMV wave [ ] 0 ψ (II): Let Ψ 1 L (R,C ) be given by Ψ 1 =. Then, for all ψ 0 f L (R,C ), we have (see Example 3.3) D Aj T k E Cj Ψ 1,f = f. j N,k Z Hence, {D Aj T k E Cj Ψ 1 } j N,k Z is a Parseval WHMV wave packet frame for L (R,C ). Similarly, {D Aj T k E Cj Ψ } j N,k Z [ is also ] a Parseval WHMV ψ 0 wave packet frame for L (R,C ), where Ψ =. 0 ψ { } Choose α 1 = α = 1. Then, α ξ W(Ψ ξ ) {D Aj T k E Cj Ψ o } j N,k Z, [ ] ψ ψ where Ψ o =. For every β > 0 and for any i {1,}, we have ψ ψ (3.6) β j N,k Z D Aj T k E Cj Ψ i,f = β f for all f L (R,C ).

15 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 167 Let 0 f L (R ) be arbitrary but fixed. Then, for 0 f o = L (R,C ), we compute (3.7) j N,k Z α ξ W(Ψ ξ ),f o = j N,k Z = j N,k Z = = 0. j N,k Z D T Aj k E Ψ Cj o,f o R [ [ ][ ] DAj T k E Cj ψ D Aj T k E Cj ψ f 0 D Aj T k E Cj ψ D Aj T k E Cj ψ f 0 ] [ ] f f 0 0 By (3.6) and (3.7), we conclude that condition (3.5) of Theorem 3.4 is not { } satisfied. Hence, the finite sum α ξ W(Ψ ξ ) is not a WHMV wave packet frame for L (R,C ). Next, we discuss optimal bounds of a WHMV wave packet frame in L (R d,c s r ). We first state and prove the following two lemmas. Lemma 3.6. Let A : L (R d,c s r ) L (R d,c s r ) be a linear operator satisfying the following conditions. 1. Af,g = f,ag for every f,g L (R d,c s r ).. traceaf,f 0 for every f L (R d,c s r ). Then, for any f,g L (R d,c s r ), (3.8) traceaf,g traceaf,f traceag,g. Proof. Let f,g L (R d,c s r ) bearbitrarybut fixed. First, we consider the case in which at least one of traceaf,f and traceag,g is non-zero. Without loss of generality, assume that traceag, g is non-zero. Then, for every scalar α, we have (3.9) 0 tracea(f+αg),f+αg = traceaf,f +α traceaf,g +α traceag,f +αα traceag,g. Choose α = traceaf,g traceag,g. Then, by using (3.9) we have This gives (3.8). 0 traceaf,f traceaf,g traceag,g.

16 168 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA Next, we discuss the case traceaf,f = 0 = traceag,g. We compute tracea(f+g),f+g tracea(f g),f g = traceaf,g +traceag,f = traceaf,g +traceg,af = traceaf,g +traceaf,g = traceaf, g + traceaf, g ( ) = 4 Real traceaf, g. Similarly, we obtain ( ) tracea(f+ig),f+ig tracea(f ig),f ig = 4 Im traceaf, g. Therefore traceaf,g = 1 ( tracea(f+g),f+g tracea(f g),f g 4 (3.10) +i tracea(f+ig),f+ig ) i tracea(f ig),f ig. Now tracea(f+g),f+g +tracea(f g),f g = traceaf,f +traceaf,g +traceag,f +traceag,g = 0. +traceaf,f traceaf,g traceag,f +traceag,g Therefore, by using condition (.), we have (3.11) Similarly, implies (3.1) tracea(f+g),f+g = 0 = tracea(f g),f g. tracea(f+ig),f+ig +tracea(f ig),f ig = 0, tracea(f+ig),f+ig = 0 = tracea(f ig),f ig. Therefore, applying (3.11) and (3.1) in (3.10), we have traceaf,g = 0 for every f,g L (R d,c s r ). Hence inequality (3.8) is proved. The proof is complete. Lemma 3.7. Let A,B : L (R d,c s r ) L (R d,c s r ) be bounded linear operators and suppose that 1. Af,g = f,ag andabf,g = f,abg for every f,g L (R d,c s r ),. traceaf,f 0 for every f L (R d,c s r ).

17 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 169 Then (3.13) traceabf,f B traceaf,f for every f L (R d,c s r ). Proof. By Lemma 3.6 and the inequality of Arithmetic and Geometric mean, we have (3.14) traceaf,g traceaf,f traceag,g 1 ( ) traceaf, f + traceag, g for any f,g L (R d,c s r ). For any positive integer n and for any f,g L (R d,c s r ), we compute (3.15) By (3.14) and (3.15), we have (3.16) AB n f,g = B n 1 f,abg = AB n 1 f,bg = B n f,ab g = = f,ab n g. traceab n f,f = tracef,ab n f = traceaf,b n f 1 [ ] traceaf,f +traceab n f,b n f = 1 [ ] traceaf,f +traceb n f,ab n f = 1 [ ] traceaf,f +traceab n f,f, for any f L (R d,c s r ). By using (3.16), for any f L (R d,c s r ), we compute (3.17) traceabf, f 1 ( ) traceaf,f +traceab f,f ( traceaf,f ( traceaf,f +traceab 4 f,f )) ( n ) traceaf,f + 1 n traceabn f,f. If B = 1, then by using Lemma.1, we have that for any f L (R d,c s r ) traceab n f,f AB n f f A f.

18 170 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA Therefore, the last term in (3.17) tends to 0 as n and we obtain traceabf, f traceaf, f for all f L (R d,c s r ). If B 1, we get (3.13) by replacing B by B/ B. Lemma 3.7 immediately yields the following proposition. Proposition 3.8. Let A,B : L (R d,c s r ) L (R d,c s r ) be bounded linear operators and suppose that Then 1. Af,g = f,ag and Bf,g = f,bg for every f,g L (R d,c s r ).. traceaf,f 0 and tracebf,f 0 for every f L (R d,c s r ). 3. AB=BA. (3.18) traceabf,f 0 for every f L (R d,c s r ). Proof. We may assume, without loss of generality, that 0 trace(i B)f,f traceif,f, where I is the identity operator on L (R d,c s r ). Since otherwise, we can replace B by βb with an appropriate positive factor β. By using Lemma. and Lemma.1, we have I B = sup trace(i B)f,f f =1,f L (R d,c s r ) sup traceif, f f =1,f L (R d,c s r ) sup f =1,f L (R d,c s r ) If f = 1. Since and A(I B) = A AB = A BA = (I B)A A(I B)f,g = (I B)f,Ag = f,(i B)Ag = f,a(i B)g for all f,g L (R d,c s r ). By Lemma 3.7, we have tracea(i B)f,f I B traceaf,f traceaf,f, for all f L (R d,c s r ). This gives traceabf,f 0 for all f L (R d,c s r ). Let {D Aj T Bk E Cm Ψ} be a WHMV wave packet frame for the function space L (R d,c s r ) with frame operator S Ψ and lower and upper

19 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 171 frame bounds L Ψ,U Ψ, respectively. Then, for any f L (R d,c s r ), we have f,s 1 Ψ D A j T Bk E Cm Ψ = U Ψ S 1 Ψ f U Ψ S 1 Ψ f. S 1 Ψ f,d A j T Bk E Cm Ψ Therefore, {S 1 Ψ D A j T Bk E Cm Ψ} is a Bessel sequence and hence, the frame operator for {S 1 Ψ D A j T Bk E Cm Ψ} is well defined. Now f,s 1 Ψ D A j T Bk E Cm Ψ S 1 Ψ D A j T Bk E Cm Ψ = S 1 Ψ = S 1 Ψ S ΨS 1 Ψ f = S 1 Ψ f, f L (R d,c s r ). S 1 Ψ f,d A j T Bk E Cm Ψ D Aj T Bk E Cm Ψ Therefore, the frame operator for {S 1 Ψ D A j T Bk E Cm Ψ} is S 1 The inequality in (3.1) can be rewritten as (3.19) L Ψ tracef,f traces Ψ f,f U Ψ tracef,f, f L (R d,c s r ). One may observe that the operators S Ψ L Ψ I and S 1 Ψ satisfy the following: 1. (S Ψ L Ψ I)f,g = f,(s Ψ L Ψ I)g and S 1 Ψ f,g = f,s 1 Ψ g, f,g L (R d,c s r ).. trace(s Ψ L Ψ I)f,f 0 (by (3.19)) and for f L (R d,c s r ), traces 1 Ψ f,f = f,s 1 Ψ D A j T Bk E Cm Ψ 0. That is (3.0) 3. (S Ψ L Ψ I)S 1 Ψ = I L ΨS 1 Ψ = S 1 Ψ (S Ψ L Ψ I). By Proposition 3.8, we have Similarly (3.1) trace(s Ψ L Ψ I)S 1 Ψ f,f 0 for all f L (R d,c s r ). traces 1 Ψ f,f L Ψ 1 tracef,f for all f L (R d,c s r ). U Ψ 1 tracef,f traces 1 Ψ f,f for all f L (R d,c s r ). Ψ.

20 17 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA By (3.0) and (3.1), for any f L (R d,c s r ), we have U Ψ 1 f f,s 1 Ψ D A j T Bk E Cm Ψ 1 LΨ f. Hence, {S 1 Ψ D A j T Bk E Cm Ψ} is frame for L (R d,c s r ) with frame bounds U 1 Ψ and L 1 Ψ. Furthermore, if L OptΨ is the optimal lower frame bound for {D Aj T Bk E Cm Ψ} and the optimal upper frame bound for {S 1 Ψ D A j T Bk E Cm Ψ} is UΨ o < L 1 Opt Ψ. Since the family {S 1 Ψ D A j T Bk E Cm Ψ} is a frame with frame operator S 1 Ψ, we get that { } {D Aj T Bk E Cm Ψ} = (S 1 Ψ ) 1 S 1 Ψ D A j T Bk E Cm Ψ has the lower frame bound (UΨ o) 1 > L OptΨ, which is a contradiction. Therefore, the family {S 1 Ψ D A j T Bk E Cm Ψ} has the optimal upper bound L 1 Opt Ψ. Similarly, U 1 Opt Ψ is the optimal lower bound for the family {S 1 Ψ D A j T Bk E Cm Ψ}. Next we observe that as in the case of ordinary frames, matrix-valued wave packet optimal frames bounds can be expressed in terms of norm of the frame operator. Indeed, by the definition of optimal upper frame bound and Lemma., we have U OptΨ = sup f =1,f L (R d,c s r ) = sup traces Ψ f,f f =1,f L (R d,c s r ) = S Ψ. D Aj T Bk E Cm Ψ,f This gives the upper optimal frame bound. Now, by part (i), L 1 Opt Ψ is the optimal upper bound for the frame {S 1 Ψ D A j T Bk E Cm Ψ} which has the frame operator S 1 Ψ. Applying the same argument as above, we obtain L 1 Opt Ψ = S 1 Ψ, that is, L Opt Ψ = S 1 Ψ 1. The next theorem gives the range } of optimal frame bounds associated with the finite sum { p α ξw(ψ ξ ). Theorem 3.9. For each ξ [p], let W(Ψ ξ ) = {D Aj T Bk E Cm Ψ ξ } be a WHMV wave packet frame for L (R d,c s r ) with the frame operator S ξ. { p } Let α 1,α,...,α p be non-zero scalars. If the finite sum α ξ W(Ψ ξ ) is a

21 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 173 frame for L (R d,c s r ) with frame operator S Ψ, then (3.) and (3.3) ( min S ξ 1 1)( ) min α ξ 1 ξ p 1 ξ p ( )( ) p(p 1) max S ξ max α ξ SΨ 1 ξ p 1 ξ p p ( )( ) max S ξ max α ξ S 1 1 ξ p 1 ξ p Ψ 1. Proof. We compute S 1 Ψ 1 f = = + p,ξ t f, f, p α ξ W(Ψ ξ ) p α ξ D Aj T Bk E Cm Ψ ξ [ p α ξ f,d Aj T Bk E Cm Ψ ξ p α ξ ] α ξ α t f,d Aj T Bk E Cm Ψ ξ f,d Aj T Bk E Cm Ψ t f,d Aj T Bk E Cm Ψ ξ p + α ξ α t f,d Aj T Bk E Cm Ψ ξ f,d Aj T Bk E Cm Ψ t,ξ t ( p p ) α ξ S ξ + α ξ α t S ξ S t f,ξ t [( ) max S p ] ξ α ξ α t f 1 ξ p [ p ( max 1 ξ p S ξ )( ) ] max α ξ f 1 ξ p for all f L (R d,c s r ). This gives (3.3).

22 174 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA To prove inequality (3.), for any f L (R d,c s r ) we compute S Ψ f p,ξ t f, p α ξ D Aj T Bk E Cm Ψ ξ [ p α ξ f,d Aj T Bk E Cm Ψ ξ ] α ξ α t f,d Aj T Bk E Cm Ψ ξ f,d Aj T Bk E Cm Ψ t p = α ξ f,d Aj T Bk E Cm Ψ ξ p α ξ α t f,d Aj T Bk E Cm Ψ ξ f,d Aj T Bk E Cm Ψ t,ξ t [( min S ξ 1 1) p ( ) p ] α ξ max S ξ α ξ α t f 1 ξ p 1 ξ p [( min S ξ 1 1)( ) min α ξ 1 ξ p 1 ξ p ( )( ) ] p(p 1) max S ξ max α ξ f. 1 ξ p 1 ξ p This gives (3.). The theorem is proved.,ξ t 4. Application to the Frame Algorithm Frame algorithms are used in approximations of signals (vectors) in the underlying space. One of the variants of classical algorithm, known as frame algorithm, which depends on the width of frames (i.e., on the knowledge of frame bounds) is given in the following theorem. Theorem 4.1 (Frame Algorithm, [4]). Let {f k } m k=1 be a frame for a finite dimensional Hilbert space V with frame bounds L, U and frame operator S. For a given f V, define {g k } k=0 V by Then (4.1) g 0 = 0, g k = g k 1 + U +L S(f g k 1) (k N). f g k ( U L ) k f (k N). U +L

23 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 175 Theorem 3. is also true for ordinary frames in a separable Hilbert space H. The following example shows that the width of the frame can be reduced by using frame bounds and scalars associated with the finite sum of frames. That is, the rate of convergence(in approximation of signals) can be increased. Example 4.. Let V = C and let {f 1 = (1,1),f = (0, ),f 3 = (,0)} C. Then, {f k } 3 k=1 is a frame for C with frame bounds L = 1, U = 4. The width of the frame {f k } 3 k=1 is given by (4.) U L U +L = 3 5 = 0.6. Consider the Parseval frame {h k } 3 k=1 {h = 1 = (1,0),h = (0,1),h 3 = } (0,0) for C. That is, frame bounds of {h k } 3 k=1 are L 1 = U 1 = 1. The finite linear sum 3 α ξf k,ξ of {f k } 3 k=1 and {h k} 3 k=1 is given by 3 ) α ξ f k,ξ = α 1 h k +α f k (where f k,1 = h k and f k, = f k, k {1,,3}. By Theorem 3., the finite sum 3 α ξf k,ξ is a frame for C with frame bounds and L o = ( α 1 L 1 α U ) = ( α 1 α ) U o = ( α 1 U 1 + α U ) = ( α 1 + α ), whenever α 1 = α 1 L 1 > α U = 4 α. Choose α 1 = 10 and α = Then, the width of the frame 3 α ξ f k,ξ is given by (4.3) U o L o U o +L o = 8 α 1 α α 1 +8 α = By (4.) and (4.3), one may observe that there is considerable decrease in the width ofthe frameassociatedwith framebounds offinite sum offrames. More precisely, the rate of convergence in the frame algorithm can be increased by using suitable scalars and frame bounds of finite sum of frames. A comparison between the width of a given frame {f k } 3 k=1 and width of finite sum of frames 3 α ξf k,ξ for five levels is illustrated in Table 1.

24 176 JYOTI, DEEPSHIKHA, L. K. VASHISHT AND G. VERMA Table 1 Level Convergence rate Convergence rate with respect to ( ) k with respect to ( ) k k= k= k= k= k= References [1] P. G. Casazza and G. Kutyniok, Finite frames. Theory and applications, Birkhäuser, 013. [] P. G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (000), [3] P. G. Casazza and R. G. Lynch, A brief introduction to Hilbert space frame theory and its applications, in Finite frame theory, AMS, 016, [4] O. Christensen, An introduction to frames and Riesz bases, Birkhäuser, 016. [5] O. Christensen and A. Rahimi, Frame properties of wave packet systems in L (R d ), Adv. Comput. Math. 9 (008), [6] A. Córdoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), [7] W. Czaja, G. Kutyniok and D. Speegle, The geometry of sets of parameters of wave packets, Appl. Comput. Harmon. Anal. 0 (006), [8] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 7 (1986), [9] I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, 199. [10] Deepshikha and L. K. Vashisht, Extension of Weyl-Heisenberg wave packet Bessel sequences to dual frames in L (R), J. Class. Anal. 8 (016), [11] Deepshikha and L. K. Vashisht, A note on discrete frames of translates in C N, TWMS J. Appl. Eng. Math. 6 (016), [1] Deepshikha and L. K. Vashisht, Necessary and sufficient conditions for discrete wavelet frames in C N, J. Geom. Phys. 117 (017), [13] Deepshikha and L. K. Vashisht, Extension of Bessel sequences to dual frames in Hilbert spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 79 (017), [14] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 7 (195), [15] Dao-Xin Ding, Generalized continuous frames constructed by using an iterated function system, J. Geom. Phys. 61 (011), [16] K. Guo and D. Labate, Some remarks on the unified characterization of reproducing systems, Collect. Math. 57 (006), [17] C. Heil, A basis theory primer, Expanded edition, Birkhäuser, 011. [18] E. Hernández, D. Labate and G. Weiss, A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal. 1 (00), [19] E. Hernández, D. Labate, G. Weiss and E. Wilson, Oversampling, quasi-affine frames, and wave packets, Appl. Comput. Harmon. Anal. 16 (004),

25 SUMS OF MATRIX-VALUED WAVE PACKET FRAMES 177 [0] Jyoti, Deepshikha, L. K. Vashisht and G. Verma, Matrix-valued wave packet frames in L (R d,c s r ), Preprint. [1] J. R. Holub, On a property of bases in a Hilbert space, Glasg. Math. J. 46 (004), [] S. K. Kaushik, G. Singh and Virender, On WH packets in L (R), Commun. Math. Appl. 3 (01), [3] D. Labate, G. Weiss and E. Wilson, An approach to the study of wave packet systems, in Wavelets, frames and operator theory, Contemp. Math. 345, 004, [4] M.Lacey and C.Thiele, L p estimates on the bilinear Hilbert transform for < p <, Ann. of Math. () 146 (1997), [5] M. Lacey and C. Thiele, On Calderón s conjecture, Ann. Math. 149 (1999), [6] S. Obeidat, S. Samarah, P. G. Casazza and J. C. Tremain, Sums of Hilbert space frames, J. Math. Anal. Appl. 351 (009), [7] K. Thirulogasanthar and W. Bahsoun, Frames built on fractal sets, J. Geom. Phys. 50 (004), [8] L. K. Vashisht and Deepshikha, Weaving properties of generalized continuous frames generated by an iterated function system, J. Geom. Phys. 110 (016), [9] S. Zhang and A. Vourdas, Analytic representation of finite quantum systems, J. Phys. A 37 (004), Jyoti Department of Mathematics University of Delhi Delhi India jyoti.sheoran3@gmail.com Deepshikha Department of Mathematics University of Delhi Delhi India dpmmehra@gmail.com L. K. Vashisht Department of Mathematics University of Delhi Delhi India lalitkvashisht@gmail.com G. Verma Centre for Industrial and Applied Mathematics School of Information Technology and Mathematical Sciences University of South Australia Adelaide Australia Geetika.Verma@unisa.edu.au Received: