2019/05/20 by Mourad Boulsane, Boulsane, M., NourElHouda Bourguiba +3 · 1 citation
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1905.08354
openalex publication_date 2019/05/20 · openalex created_date 2019/05/29 · openalex updated_date 2026/07/28
For fixed W∈ (0,(1)/(2)) and positive integer N≥ 1, the discrete prolate spheroidal wave functions (DPSWFs), denoted by Uk,WN, 0≤ k≤ N-1 form the set of the eigenfunctions of the positive and finite rank integral operator \widetilde QN,W, defined on L2(-1/2,1/2), with kernel KN(x,y)=(sin(Nπ(x-y)))/(sin(π(x-y))) \mathbf 1[-W,W](y). It is well known that the DPSWF's have a wide range of classical as well as recent signal processing applications. These applications rely heavily on the properties of the DPSWFs as well as the behaviour of their eigenvalues \widetilde λk,N(W). In his pioneer work \citeSlepian, D. Slepian has given the properties of the DPSWFs, their asymptotic approximations as well as the asymptotic behaviour and asymptotic decay rate of these eigenvalues. In this work, we give further properties as well as new non-asymptotic decay rates of the spectrum of the operator \widetilde QN,W. In particular, we show that each eigenvalue \widetilde λk,N(W) is up to a small constant bounded above by the corresponding eigenvalue, associated with the classical prolate spheroidal wave functions (PSWFs). Then, based on the well established results concerning the distribution and the decay rates of the eigenvalues associated with the PSWFs, we extend these results to the eigenvalues \widetilde λk,N(W). Also, we show that the DPSWFs can be used for the approximation of classical band-limited functions and they are well adapted for the approximation of functions from periodic Sobolev spaces. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.