10_445_kumar.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 4, 2010, 717-724 ISSN 1307-5543 – www.ejpam.com Prolate Spheroidal Wavelet Coefficients,Frames and Double Infinite Matrices Devendra Kumar Department of Mathematics [Research and PostGraduate Studies], M.M.H.College,Model Town,Ghaziabad- 201001,U.P.India Abstract. In this paper we defined the double infinite matrix A= a(m, n, k)and study the action of A on f ∈ L2(R) and on its prolate spheroidal wavelet coefficients. We also find the frame condition for A-transform of f ∈ L2(R) whose wavelet series expansion is known. 2000 Mathematics Subject Classifications: 42C15,41A17,42C40 Key Words and Phrases: Prolate spheroidal wave functions,double infinite matrix and frame 1. Introduction The study of continuous prolate spheroidal wave functions (PSWFs) has been an active area of research in both electrical engineering and mathematics. Yet they seem to be an in- exhaustible and inspirational source of new ideas and methods, both theoretical and applied. The PSWFs are those that are most highly,localized simultaneously in both the time and fre- quency domain. This fact was discovered by Slepian and his collaborators and was presented in a series of articles [7],[8],[12]-[14] about forty years ago. Let us recall the connection between PSWFs and the Shannon sampling theorem (Shan- non[10]) given by the formula f (t) = ∞ ∑ n=−∞ f (n) sinπ(t − n) π(t − n) . (1) The above formula (1) holds for π-bandlimited signals with finite energy,that is,for continuous functions in L2(R) whose Fourier transform has support in [−π,π]. This theorem has became a well known part of both the mathematical and engineering literature. The sinc function S(t) = sinπt πt which appears in this formula is closely related to the PSWFs ϕn,σ,τ(t). Email address: d_kumar001�rediffmail. om http://www.ejpam.com 717 c© 2010 EJPAM All rights reserved. D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 717-724 718 The PSWFs ϕn,σ,τ(t) constitute an orthogonal basis of the space of σ-band limited func- tions on the real line. They are maximally concentrated on the interval [−τ,τ] and depend on parameters σ and τ. PSWFs are characterized as the eigenfunctions of an integral operator with kernel arising from the sinc functions S(t) : σ π ∫ τ −τ ϕn,σ,τ(x)S( σ π (t − x))d x = λn,σ,τϕn,σ,τ(t), |t| ≤ τ. (2) It is easy to show that the symmetrical kernel S(σ π (t − x)) is positive definite,so that from[1] we know that (2) has solutions in L2(−τ,τ) only for a discrete set of real positive values of λn,σ,τ say λ0,σ,τ ≥ λ1,σ,τ ≥ . . . and that as n → ∞, limλn,σ,τ = 0. The corresponding solutions,or eigenfunctions,ϕ0,σ,τ(t),ϕ1,σ,τ(t), . . . can be chosen to be the real and orthogonal on (−τ,τ). The variational problem that let to (2) only requires that equation to hold for |t| ≤ τ. With ϕn,σ,τ(x) on the left of (2) gives for |x | ≤ τ, however, the left is well defined for all t. We use this to extend the range of definition of the ϕn,σ,τ’s and so define ϕn,σ,τ(t) = σ πλn,σ,τ ∫ τ −τ ϕn,σ,τ(x)S( σ π (t − x))d x , |t|> τ. The eigenfunctions ϕn,σ,τ are now defined for all t. In addition to the equation (2), the {ϕn,σ,τ} satisfy an integral equation over (−∞,∞) σ π ∫ ∞ −∞ ϕn,σ,τ(x)S( σ π (t − x))d x = (ϕn,σ,τ ∗ Sσ)(t) = ϕn,σ,τ(t) with the same kernel. This leads to a dual orthogonality ∫ τ −τ ϕn,σ,τ(x)ϕm,σ,τ(x)d x = λn,σ,τδnm, ∫ ∞ −∞ ϕn,σ,τ(x)ϕm,σ,τ(x)d x = δnm and the fact that they constitute an orthogonal basis of L2(−τ,τ), as well as an orthonormal basis of the subspaces Bσ of L2(−∞,∞), the Paley-Wiener space of all σ−bandlimited func- tions. We are interested mainly in ϕ0,σ,τ whose concentration on the interval [−τ,τ]is maxi- mum. Since ϕk,σ,τ has exactly k zeros in the interval [−τ,τ],so ϕ0,σ,τ (PSWFs)are entire functions and therefore can not vanish on any interval,they can be made uniformly small outside of [−τ,τ] for τ or σ sufficiently large, so that computationally they behave like func- tions with compact support. To construct PS wavelets,the scaling function φ = ϕ0,π,τ, where τ is any positive number,was introduced by [15] and obtained a basis composed of a space V0 ⊂ L2 (R) which turns out to be the Paley-Wiener space Bπ of π - bandlimited functions. D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 717-724 719 A multiresolution analysis (MRA)are then based on this construction. The other spaces are obtained by dilation by factors of two and consist of the Paley-Wiener spaces Vm=B2mπ.The sinc function is the standard scaling function of this MRA. It is well known that sinc function has very good frequency localization, but not very good time localization. It follows that this wavelet basis has limited use in comparison to the Daubechies wavelets which have compact support in the time domain. However,PSWFs are superior to sinc function and they are similar to the Daubechies wavelets for practical computations. Using the standard wavelet approach in which dilations of ϕ0,π,τ(2 mt) are used to get the basis {ϕ0,π,τ(2 mt − n)} of Vm we get φ(2m t) = ϕ0,π,τ(2 mt) = 2m/2ϕ0,2mπ,2−mπτ(t), which show that the concentration interval becomes smaller as m increases. So we have to fix the concentration interval by taking {ϕ0,2mπ,τ(t − 2−mn)} instead as a possible Riesz basis of Vm. Thus our new basis for V0 and Vm are different from the standard wavelet basis for V0 and Vm consisting of translates of the sinc function. 2. Frames and Applications to Prolate Spheroidal Wavelets The notion of frame goes back to Duffin and Schaeffer[6] in the early 1950s to deal with the problems in nonharmonic Fourier series. In many cases the wavelet experts prefer to work with frames instead of Riesz bases. The recent development and work on frames and related topics,(see[2][3][4]). In this paper,we will use the double infinite matrices (see [9][10]) to obtain the frame conditions and prolate spheroidal wavelet coefficients. A sequence {xn} in a Hilbert space H is a frame if there exist constants c1 and c2, 0< c1 ≤ c2 <∞, such that c1|| f || 2 ≤ ∑ n∈z |〈 f , xn〉| 2 ≤ c2|| f || 2, (3) for all f ∈ H. The spermium of all such numbers c1 and infimum of all such numbers c2 are called the frame bounds of the frame. The frame is called tight frame when c1 = c2 and is called normalized tight frame when c1 = c2 = 1. Any orthonormal basis in a Hilbert space H is a normalized tight frame. In another paper[5] we have proved that a prolate spheroidal wavelet function φm(t − 2−mn) = ϕ0,2mπ,τ(t − 2−mn) ∈ L2(R), constitute a frame with frame bounds c1 and c2, if any f ∈ L2(R) such that c1|| f || 2 ≤ ∞ ∑ m,n=−∞ |〈 f ,φm(t − 2−mn)〉|2 ≤ c2|| f || 2. Let A = a(m, n,κ) = ∫∞ −∞ φm(t − 2−mn)φm(t − 2−mκ)d t be a double infinite matrix of real numbers. Then, A-transform of a double sequence {φκm} is defined as Aφκm = ∫ ∞ −∞ φm(t − 2−mn)φm(t − 2−mκ)φκmd t D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 717-724 720 which is called A-means of the sequenceφκm. This definition is due to Moricz and Rhoades [9]. A double matrix A= a(m, n,κ) is satisfied the following conditions : (i) limm→∞ a(m, n,κ) = 1, n,κ ∈ z (ii) ||A||= supm>0 |a(m, n,κ)| <∞, n,κ ∈ z. The approximation of a function in L2(R) by function in Vm is given by a series of the form f (t) = ∑ κ bm κ φm(t − 2−mκ) (4) where the coefficient may be obtained from the dual Riesz basis. In this case the result is the projection of a function f onto Vm. The coefficients for the projection are bm κ = ∫ ∞ −∞ f (t)φ̃m(t − 2−mκ)d t. The kernel of this projection is given by qm(x , t) = ∑ κ φm(x − 2−mκ)φ̃m(t − 2−mκ), (5) here φ̃m is biorthogonal to φm. 3. Main Results In this section we prove the following theorems. Theorem 1. Let A= a(m, n,κ be a double infinite matrix. If f (t) = ∑ κ bm κ φm(t − 2−mκ) is a PS wavelet expansion of f ∈ L2(R) with wavelet coefficients bm κ = ∫ ∞ −∞ f (t)φ̃m(t − 2−mκ)d t = 〈 f , φ̃m(t − 2−mκ)〉, then the frame condition for A-transform of f ∈ L2(R) is c1|| f || 2 2 ≤ ∑ κ |〈Af ,qm(y, t)〉|2 ≤ c2|| f || 2 2 (6) where qm is given by (5), Af is the A-transform of f and 0< c1 ≤ c2 <∞. D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 717-724 721 Proof. By (3), we can write f (t) = ∑ κ bm κ φm(t − 2−mκ) taking A-transform of f , we get Af = ∫ ∞ −∞ φm(t − 2−mn)φm(t − 2−mκ)Σκbm κφm(t − 2−mκ)d t = Σκ ∫ ∞ −∞ φm(t − 2−mn)φm(t − 2−mκ) ∫ ∞ −∞ f (y)φ̃m(y − 2−mκ)φm(t − 2−mκ)d yd t = Σκ〈Af , φ̃m(y − 2−mκ)〉φm(t − 2−mκ) = 〈Af ,Σκφ̃m(y − 2−mκ)φm(t − 2−mκ)〉 = 〈Af ,qm(y, t)〉, therefore Σκ|〈Af ,qm(y, t)〉|2 ≤ Σκ ∫ ∞ −∞ |Af qm(y, t)|2d t ≤ Σκ ∫ ∞ −∞ (Af )2d t ∫ ∞ −∞ [S2m(y − t)]2d t = ||A||22|| f || 2 2 (7) (since qm(y, t) = sin2mπ(y−t) π(y−t) = S2m(y − t)) Now, for any arbitrary f ∈ L2(R), define f̃ = [Σκ|〈Af ,qm(y, t)〉|2]−1/2 f . Clearly 〈Af̃ ,qm(y, t)〉 = [Σκ|〈Af ,qm(y, t)〉|2]−1/2〈Af ,qm(y, t)〉 then Σκ|〈Af ,qm(y, t)〉|2 ≤ 1. If there exist a positive constant α, then ||Af̃ ||22 ≤ α, so [Σκ|〈Af ,qm(y, t)〉|2]−1||Af ||22 ≤ α or [Σκ|〈Af ,qm(y, t)〉|2]−1|〈Af ,qm(y, t)〉|2 ≤ α ≤ [Σκ|〈Af ,qm(y, t)〉|2]−1||A||22|| f || 2 2 ≤ α D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 717-724 722 or c1|| f || 2 2 ≤ Σκ|〈Af ,qm(y, t)〉|2. (8) Combining (7) and (8) we get c1|| f || 2 2 ≤ Σκ|〈Af ,qm(y, t)〉|2 ≤ c2|| f || 2 2. Hence the proof is completed. Theorem 2. If bm κ are the PS wavelet coefficients of f ∈ L2(R), that is, bm κ = 〈 f , φ̃m(t − 2−m)〉 then the dm = 〈 f ,qm(y, t)〉, where {dm} is defined as the A-transform of {bm κ } by dm = ∑ n,κ a(m, n,κ)bm κ φm(t − 2−mκ). (9) Proof. We have ∑ n,κ a(m, n,κ)bm κ φm(t − 2−mκ) = ∑ n,κ 〈φm(t − 2−mn),φm(t − 2−mκ)〉〈 f , φ̃m(t − 2−mn)〉φm(t − 2−mn) = ∫ ∞ −∞ ∑ n,κ φm(t − 2−mn)φm(t − 2−mκ)d t ×( ∫ ∞ −∞ f (y)φ̃m(y − 2−mκ)d y)φm(t − 2−mκ) = ∫ ∞ −∞ ∑ n,κ φm(t − 2−mn)φ̃m(y − 2−mκ)(φm(t − 2−mn))2d t ∫ ∞ −∞ f (y)d y = ∫ ∞ −∞ qm(y, t) f (y)d y ∫ ∞ −∞ [φm(t − 2−mn)]2d t = 〈 f ,qm(y, t)〉. Hence the proof is completed. Theorem 3. Let A= a(m, n,κ) be a double nonnegative infinite matrix then {qm(y, t)} constitute a frame of L2(R). Proof. We have ∑ m |dm| 2 = ∑ m |〈 f ,qm(y, t)〉|2 = 1 2π ∑ m |〈 f̂ , q̂m(y, t)〉|2 REFERENCES 723 = 1 2π ∑ m ∫ ∞ −∞ | 1 2π ∫ ∞ −∞ f̂ (ξ), q̂m(w,ξ)dξ|2dw = 1 2π ∑ m ∫ ∞ −∞ |χ2mπ(w) f̂ (w)| 2dw = 1 2π ∑ m ∫ 2mπ −2mπ | f̂ (w)|2dw = 1 2π ∫ ∞ −∞ | f̂ (w)|2dw = || f ||22 that is, ∑ m |dm| 2 = || f ||22, f ∈ L2(R). Therefore, for matrix A=a(m,n,κ), we have c1|| f || 2 2 ≤ ∑ m |dm| 2 ≤ c2|| f || 2 2, where 0≤ c1, c2 <∞. This completes the proof of the theorem. References [1] R. Courant and D. Hilbert , Methods of Mathematical Physics , Vol 1 , Interscience Pub- lishers , New York, pp 122-134. 1955. [2] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41, No -7, 909-996. 1988. [3] I. Daubechies , The wavelet transform , time frequency localization and signal analysis, IEEE Trans. Inform. Theory 36, No- 5, 961-1005. 1990. [4] I. Daubechies , Ten Lectures on Wavelets, CBMS - NSF Regional Conference Series in Applied Mathematics , Vol 61, Society for Industrial and Applied Mathematics (SIAM) Pennsylvania, 1992. [5] Devendra Kumar, Convergence of prolate spheroidal wavelets in a generalized sobolev space and frames, Submitted for publication in Indian Journal of Industrial and Applied Mathematics. [6] R.J. Duffin and A.C. Schaeffer, A calss of nonharmonic Fourir series , Trans. Amer. Math. Soc. 72, 341-366. 1952. REFERENCES 724 [7] H.J. Landau and H.O. Pollak , Prolate sheroidal wave functions, Fourier analysis and uncertainity , II , Bell System Tech. J. 40. 65-84. 1961. [8] H.J. Landau and H.O. Pollak , Prolate sheroidal wave functions, Fourier analysis and uncertainity , III , Bell System Tech. J. 41. 1295-1336. 1982. [9] F. Moricz and B.E. Rhaodes, Comparison theorems for double summability methods, Publ. Math. Debrecen 36, No 1-4, 207- 220. 1989. [10] G.M. Robinson, Divergent double sequences and series, Trans. Amer. Math. Soc. 28, 50- 73. 1926. [11] C.E. Shannon, Communication in the presence of noise , Proc. IRE 37, 10-21. 1949. [12] D. Slepian and H.O. Pollak , Prolate spheroidal wave functions, Fourier analysis and uncertainity , I. Bell System Tech. J. 40, 43-64. 1961 [13] D. Slepian , Prolate spheroidal wave functions, Fourier analysis and uncertainity , IV Bell System Tech. J. 43, 3009-3058. 1964. [14] D. Slepian, Some comments on Fourier analysis , uncertainity and modeling , SIAM Review, 25, 379-393. 1983. [15] G.G. Walter and Xiaoping Shen, Wavelet based on prolate spheroidal wave functions,The J.Fourier Analysis and Applications 10, Issue 1, 1-26. 2004.