MATRIX TRANSFORM OF IRREGULAR WEYL-HEISENBERG WAVE PACKET FRAMES FOR L 2 (R)

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1 TWMS J. App. Eng. Math. V.7, N.2, 2017, pp MATIX TANSFOM OF IEGULA WEYL-HEISENBEG WAVE PACKET FAMES FO L 2 (). KUMA 1, ASHOK K. SAH 2, Astract. Cordoa and Fefferman [4] 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 this paper, we introduce the concept of matrix transform M = (α p,q,r,j,k,m ) and with the help of matrix transform we study the action of M on f L 2 () and on its wave packet coefficients. Further, we also otain the tight frame condition for matrix transform of f L 2 () whose wave packet series expansion is known. Keywords: frames, matrix transform, wave packet system. AMS Suject Classification: Primary 42C15; Secondary 42C30, 42B Introduction and Preliminaries The theory of frames for Hilert spaces were formally introduced y Duffin and Schaeffer [6], to deal with nonharmonic Fourier series. A system {f k } in a separale Hilert space H with inner product.,. is called frame (Hilert) for H if there exists positive constants A and B such that A f 2 f, f k 2 l 2 B f 2, for all f H. (1) If upper inequality in (1) holds, then we say that {f k } is a Bessel sequence for H. The positive constants A and B are called lower and upper frame ounds of the frame {f n }, respectively. They are not unique. The positive constants, A 0 = sup{a : A satisfy (1)} B 0 = inf{b : B satisfy (1)} are called optimal or est ounds of the frame. A frame {f n } for H is called tight if it is possile to choose A = B and normalized tight if A = B = 1. If removal of one f j renders the collection {f k } no longer a frame for H, then {f k } is called an exact frame for H. The operator V : l 2 H given y V ({c k }) = c k f k, {c k } l 2 k=1 1 Department of Mathematics, Kirori Mal College, University of Delhi, Delhi , India. rajkmc@gmail.com, OCID: 2 Department of Mathematics, University of Delhi, Delhi , India. ashokmaths2010@gmail.com, OCID: Manuscript received: July 10, 2016; accepted: Octoer 12, TWMS Journal of Applied and Engineering Mathematics Vol.7, No.2; c Işık University, Department of Mathematics, 2017; all rights reserved. 200

2 .KUMA, A.K.SAH: MATIX TANSFOM OF IWH WAVE PACKET FAMES FO L 2 () 201 is called the synthesis operator or pre-frame operator of the frame. Adjoint of V is the operator V : H l 2 given y V (f) = { f, f k } and is called the analysis operator of the frame {f k }. Composing V and V we otain the frame operator S = V V : H H given y S(f) = f, f k f k, f H. k=1 The frame operator S is a positive continuous invertile linear operator from H onto H. Every vector f H can e written as: f = SS f = S f, f k f k. (econstruction formula) (2) k=1 The series given in (2) converges unconditionally for all f H and is called frame decomposition or reconstruction formula for the frame. Thus, frames are redundant uilding locks which have asis like properties. One may oserve that the frame decomposition shows that all information aout a given signal (vector) f H is contained in the system { S f, f k }. The scalars S f, f k are called frame coefficients. Today frame theory is a central tool in applied mathematics and engineering. In particular frames are widely used in sampling theory, wavelet theory, wireless communication, signal processing, image processing, differential equations, filter anks, geophysics, quantum computing, wireless sensor network, multiple-antenna code design and many more. An introduction to frames in applied mathematics and engineering can e found in a so nice ook y Casazza and Kutynoik [1]. In the theoretical direction, powerful tools from operator theory and Banach spaces are eing employed to study frames. For asic theory in frames, an interested reader may e refer to [3, 4, 7, 11]. The wave packet systems were introduced and studied y Cordoa and Fefferman [4] y applying certain collections of dilations, modulations and translations to the Gaussian function in the study of some classes of singular integral operators. Leate et al. [9] adopted the same expression to descrie any collections of functions which are otained y applying the same operations to a finite family of functions in L 2 (). More precisely, Gaor systems, wavelet systems and the Fourier transform of wavelet systems are special cases of wave packet systems. Wave packet systems have recently een successfully applied to some prolems in harmonic analysis and operator theory. The wave packet systems have een studied y several authors, see [2, 5, 8, 10]. Now, we give notations and definitions required in this paper. We now recall asic notations and definitions. For 1 p <, let L p () denote the Banach space of complex-valued Leesgue integrale functions f on satisfying ( ) 1 f p = f(t) p p dt <. For p = 2, an inner product on L p () is given y f, g = fgdt, where g denotes the complex conjugate of g. We consider the unitary operators on L 2 () which are given y : Translation T a f(t) = f(t a), a.

3 202 TWMS J. APP. ENG. MATH. V.7, N.2, 2017 Modulation E f(t) = e 2πit f(t),. Dilation D a f(t) = 1 a f( t a ), a, a 0. For a > 0,, c and f L2 (), it is easy to verify that (D aj fˆ) = D a j ˆf, (E fˆ) = T ˆf, (Tc fˆ) = E c ˆf, (D aj T k E cm fˆ) = D a E k T cm ˆf. j 2. Matrix transform of IWH wave packet frames for L 2 () Definition 2.1. Let ψ L 2 (), {a j } j Z +, {c m } m Z and 0. A system of the form {D aj T k E cm ψ} is called an irregular Weyl-Heisenerg wave packet system (or IW H wave packet system). Definition 2.2. If IW H wave packet system {D aj T k E cm ψ} constitutes a frame for L 2 (), i.e., if there exists positive constants a 0 and 0 such that a 0 f 2 f, D aj T k E cm ψ 2 0 f 2, for all f L 2 (), (3) then we say that {D aj T k E cm ψ} j,k Z is an irregular Weyl-Heisenerg wave packet frame (or IW H wave packet frame). The positive constants a 0 and 0 are called lower and upper frame ounds of the irregular Weyl-Heisenerg wave packet frame {D aj T k E cm ψ}, respectively. If upper inequality in (3) is satisfied, then {D aj T k E cm ψ} is called the wave packet Bessel sequence for L 2 () with Bessel ound 0. Example 2.1. Let ψ = χ [0,1] and let a j = 2 j, j Λ = {1, 2,..., n}. Choose = 1 and c m = m, for all m Z. Then, for f L 2 () we have f, D aj T k E cm ψ 2 = f, D 2 jt k E m ψ 2 = = f, D 2 jt k E m χ [0,1] 2 D 2 jf, T k E m χ [0,1] 2. Now, {E m T k χ [0,1] } m,k Z is an orthonormal asis for L 2 () (see [3], p. 71). Therefore, D 2 jf, T k E m χ [0,1] 2 = D 2 jf 2 = n f 2. j Λ Hence, { D aj T k E cm ψ } is a wave packet frame for L2 (). Example 2.2. Let ψ = χ [0,1] and let a j = 2 j, j Z. Choose = 1 and c m = m, for all m Z. Then, { D aj T k E cm ψ } is not a wave packet frame for L2 ().

4 .KUMA, A.K.SAH: MATIX TANSFOM OF IWH WAVE PACKET FAMES FO L 2 () 203 Indeed, choose f o = χ [0,1], we compute f o, D aj T k E cm ψ 2 = f o, D 2 jt k E m ψ 2 = f o, D 2 jt k E m χ [0,1] 2 = D 2 jf o, T k E m χ [0,1] 2 = j Z D 2 jf o 2 > n f o 2, for any n. Hence, { D aj T k E cm ψ } does not satisfy the upper frame condition for L2 (). For any function ψ L 2 (), we consider the system of functions {ψ j,k,m } L 2 () as {ψ j,k,m (x) := D aj T k E cm ψ(x) : j, k, m Z, x } (4) By taking Fourier transform to (4) we otain ˆψ j,k,m (ξ) = a /2 j Thus, y Plancheral theorem, we have c j,k,m = f, ψ j,k,m = ˆψ(a j ξ c m )e 2πika j ξ. f(x)ψ j,k,m (x)dx, f L 2 () (5) We say that the system defined in (4) constitutes a wave packet frame for L 2 () if there exists positive constants C and D such that C f 2 f, ψ j,k,m 2 D f 2, for all f L 2 (). The constants C and D are known respectively as lower and upper frame ounds. If C = D, then we call it tight frame. It is called Parseval frame if C = D = 1 and in this case, every function f L 2 () can e written as f(x) = c j,k,m ψ j,k,m (x) (6) where c j,k,m = f, ψ j,k,m are given y (5), and we say wave packet co-efficient of the series (6). Definition 2.3. Let M = (α p,q,r,j,k,m ) e an infinite matrix of real numers. Then M- transform of {x j,k,m } is defined as α p,q,r,j,k,mx j,k,m. The following theorem gives sufficient condition for the matrix transform of the sequences of wave packet coefficients to e elongs to C 0. Theorem 2.1. Let M = (α p,q,r,j,k,m ) e an infinite matrix whose elements are of the form α p,q,r,j,k,m = ψ p,q,r, ψ j,k,m and if (i) j,k,m ψ j,k,m f(y)ψ j,k,m(y)dy = 1 (ii) lim p,q,r ψ p,q,r (x) = 0. Then, M-transform of the sequence of wave packet coefficients {c j,k,m } elong to C 0.

5 204 TWMS J. APP. ENG. MATH. V.7, N.2, 2017 Proof. Since, the elements of an infinite matrix of the type ψ j,k,m, ψ p,q,r and the wave packet coefficients {c j,k,m } are given y equation (5). We have Therefore, α p,q,r,j,k,m c j,k,m = ψ j,k,m, ψ p,q,r f, ψ j,k,m = ψ j,k,m (x)ψ p,q,r (x)dx f(x)ψ j,k,m (x)dx = f(x)ψ p,q,r (x)dx ψ j,k,m (x)ψ j,k,m (x)dx. α p,q,r,j,k,m c j,k,m = Using conditions (i) and (ii), we have lim α p,q,r,j,k,m c j,k,m = p,q,r This completes the proof. ψ p,q,r (x)ψ j,k,m (x)f(y)ψ j,k,m (y)dxdy. lim ψ p,q,r (x)dx = 0. p,q,r The following theorem gives sufficient condition for M-transform of f L 2 () to e satisfy frames condition. Theorem 2.2. Let M = (α p,q,r,j,k,m ) e a non-negative infinite matrix with ψ p,q,r 2 = 1 and if c j,k,m are the wave packet coefficient associated with the wave packet expansion (6). Then, the frame condition for M-transform of f L 2 () is given y C ψ f 2 Mf, ψ p,q,r 2 D ψ f 2, Where Mf is the M-transform of f L 2 () and 0 < C ψ D ψ <. Proof. We have, f(x) = f, ψ j,k,m ψ j,k,m (x). Taking M-transform of f, we get Mf(x) = Mf, ψ p,q,r ψ p,q,r (x). Therefore, Thus, we have Mf, ψ p,q,r 2 where D ψ is a positive constant. For any, f L 2 (), we define g(x) = M 2 f 2 Mf(x) 2 ψ p,q,r (x) 2 dx ψ p,q,r 2. Mf, ψ p,q,r 2 D ψ f 2, (7) Mf, ψ p,q,r f(x).

6 .KUMA, A.K.SAH: MATIX TANSFOM OF IWH WAVE PACKET FAMES FO L 2 () 205 Clearly, Hence, Mg, ψ p,q,r = Mf, ψ p,q,r Mf, ψ p,q,r 2 1. Mf, ψ p,q,r. Now, if there exits a constant K > 0 such that Mg 2 K, then Mf, ψ p,q,r 2 f 2 M A 2 = C ψ > 0. Thus, Using (7) and (8), we have This completes the proof. C ψ f 2 C ψ f 2 2 Mf, ψ p,q,r 2. (8) Mf, ψ p,q,r 2 D ψ f 2. The following theorem gives matrix transform of wave packet coefficients in terms of series. Theorem 2.3. If {c j,k,m } are the wave packet coefficients of f L 2 () with ψ p,q,r = 1. Then d p,q,r = α p,q,r,j,k,m c j,k,m, where, {d p,q,r } is defined as the M-transform of c j,k,m. Proof. By taking M-transform of equation (5), we have Mf, ψ j,k,m = Mf(x)ψ j,k,m (x)dx = α p,q,r,j,k,m c j,k,m ψ j,k,m (x)ψ p,q,r (x)dx. Thus, Mf, ψ p,q,r ψ p,q,r (x) = = = α p,q,r,j,k,m c j,k,m ψ p,q,r (x) ψ p,q,r (x) 2 dx d p,q,r ψ p,q,r (x) ψ p,q,r (x) 2 dx d p,q,r ψ p,q,r (x). Hence, d p,q,r = Mf, ψ p,q,r are the wave packet coefficients of Mf.

7 206 TWMS J. APP. ENG. MATH. V.7, N.2, 2017 Finally, the authors have given sufficient condition for the sequences {ψ j,k,m } to e a tight wave packet frames for L 2 () in Theorem 2.4. Theorem 2.4. Let M = (α p,q,r,j,k,m ) e an infinite matrix whose elements are ψ j,k,m, ψ p,q,r and let ˆψ(a p ξ c r ) 2 = 1, a.e. ξ. Then d p,q,r 2 = p,q,r f, ψ j,k,m 2 = 1 f 2. where d p,q,r = f, ψ p,q,r is the M-transform of the wave packet coefficients c j,k,m. Proof. We compute, α p,q,r,j,k,m c j,k,m = ψ j,k,m, ψ p,q,r f, ψ j,k,m = ψ j,k,m (x)ψ p,q,r (x)dx f(x)ψ j,k,m (x)dx = f(x)ψ p,q,r (x)dx ψ j,k,m (x) 2 dx = f(x)ψ p,q,r (x)dx = f, ψ p,q,r = d p,q,r. Therefore, d p,q,r 2 = = = = = = p Z = p Z = 1 a p 1 α p,q,r,j,k,m c j,k,m 2 f, ψ p,q,r 2 ˆf, ˆψ p,q,r 2 ˆf, D a p E qt cr ˆψ 2 ˆf, E qa D p a T 2 c p r ˆψ r Z q Z ap r Z 0 ap 0 ˆf(ξ) l Z ( l Z ˆψ(a p ξ c r )e 2πiqa p ξ dξ 2 ˆf(ξ a pl ) ˆψ(a p ξ l c r)dξ 2 ) ( ˆf(ξ a pl ) ˆψ(a p ξ l c r) l Z ˆf(ξ a pl ) ) ˆψ(a p ξ l c r) dξ.

8 .KUMA, A.K.SAH: MATIX TANSFOM OF IWH WAVE PACKET FAMES FO L 2 () 207 Let F (ξ) = ˆf(ξ l Z apl ) ˆψ(a p ξ l c r). Therefore, d p,q,r 2 = f, ψ p,q,r 2 = 1 = 1 = 1 ap = 1 = 1 = 1 f 2. This completes the proof. 0 l Z ˆf(ξ) ( l Z ˆf(ξ) ˆf(ξ) 2 ˆf(ξ a pl ˆψ(a p ξ c r )F (ξ)dξ ˆf(ξ) ˆψ(a ˆf(ξ) ˆψ(a p ξ c r ) ) ˆψ(a p ξ l c r)f (ξ)dξ p ξ c r ) ˆf(ξ a pl ˆψ(a p ξ c r ) 2 dξ ˆψ(a p ξ c r )dξ ) ) ˆψ(a p ξ l c r)dξ Acknowledgement The authors would like to express their sincere thanks to anonymous referee for valuale constructive comments, suggestions which improved this paper. eferences [1] Casazza,P.G. and Kutyniok,G., (2012), Finite frames: Theory and Applications, Birkhäuser. [2] Christensen,O. and ahimi,a., (2008), Frame properties of wave packet systems in L 2 (), Adv. Comput. Math., 29, pp [3] Christensen,O., (2008), Frames and Bases, An Introductory Course, Birkhäuser, Boston. [4] Cordoa,A. and Fefferman,C., (1978), Wave packets and Fourier integral operators, Comm. Partial Differential Equations, 3, 11, pp [5] Czaja,W., Kutyniok,G., and Speegle,D., (2006), The Geometry of sets of prameters of wave packets, Appl. Comput. Harmon. Anal., 20, pp [6] Duffin,.J. and Schaeffer,A.C., (1952), A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72, pp [7] Heil,C., (2011), A Basis Theory Primer, Expanded edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York. [8] Hernández,E., Laate,D., Weiss,G.,and Wilson,E., (2004), Oversampling, quasi-affine frames and wave packets, Appl. Comput. Harmon. Anal., 16, pp [9] Laate,D., Weiss,G.,and Wilson,E., (2004), An approach to the study of wave packet systems, Contemp. Math., 345, pp [10] Sah,A.K.,and Vashisht,L.K., (2015), Irregular Weyl-Heisenerg wave packet frames in L 2 (), Bull. Sci. Math., 139, pp [11] Young,., (2001), An introduction to nonharmonic Fourier series, Academic Press, New York.

208 TWMS J. APP. ENG. MATH. V.7, N.2, 2017 aj Kumar got his M.Sc. degree in Mathematics from IIT Delhi, Delhi. He received his Ph.D. degree in mathematics from University of Delhi, Delhi.

He has given talks at national/international workshops/conferences. His area of research is Theory of Frames for Hilert and Banach spaces and Wavelets. Ashok Kumar Sah graduated from L. S. College, B.

9 208 TWMS J. APP. ENG. MATH. V.7, N.2, 2017 aj Kumar got his M.Sc. degree in Mathematics from IIT Delhi, Delhi. He received his Ph.D. degree in mathematics from University of Delhi, Delhi. Currently, he is a senior assistant professor in the Department of Mathematics at the Kirori Mal College, University of Delhi, Delhi. He has given talks at national/international workshops/conferences. His area of research is Theory of Frames for Hilert and Banach spaces and Wavelets. Ashok Kumar Sah graduated from L. S. College, B..A. Bihar University, Muzaffarpur, in He got his M.Sc. degree in mathematics from University of Delhi, Delhi, in He received his M.Phil degree in mathematics from University of Delhi, Delhi, in Currently, he is an assistant professor in the Department of Mathematics at the Kirori Mal College, University of Delhi, Delhi. He has given talks at national and international workshops/conferences. His area of research are Theory of Frames, wavelets, signal processing.

WEYL-HEISENBERG FRAMES FOR SUBSPACES OF L 2 (R)

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Numer 1, Pages 145 154 S 0002-9939(00)05731-2 Article electronically pulished on July 27, 2000 WEYL-HEISENBERG FRAMES FOR SUBSPACES OF L 2 (R)