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• W2945964984 abstract "For fixed $Win big(0,frac{1}{2}big)$ and positive integer $Ngeq 1,$ the discrete prolate spheroidal wave functions (DPSWFs), denoted by $U_{k,W}^N,$ $0leq kleq N-1$ form the set of the eigenfunctions of the positive and finite rank integral operator $widetilde Q_{N,W},$ defined on $L^2(-1/2,1/2),$ with kernel $K_N(x,y)=frac{sin(Npi(x-y))}{sin(pi(x-y))}, mathbf 1_{[-W,W]}(y).$ 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 $widetilde lambda_{k,N}(W).$ In his pioneer work cite{Slepian}, 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 $widetilde Q_{N,W}.$ In particular, we show that each eigenvalue $widetilde lambda_{k,N}(W)$ 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 $widetilde lambda_{k,N}(W)$. 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." @default.
• W2945964984 created "2019-05-29" @default.
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• W2945964984 date "2019-05-20" @default.
• W2945964984 modified "2023-10-06" @default.
• W2945964984 title "Discrete Prolate Spheroidal Wave Functions: Further spectral analysis and some related applications" @default.
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• W2945964984 doi "https://doi.org/10.48550/arxiv.1905.08354" @default.
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