For fixed W ∈ 0, 1 2 and positive integer N ≥ 1, the discrete prolate spheroidal wave functions (DPSWFs), denoted by U N k,W , 0 ≤ k ≤ N − 1 form the set of the eigenfunctions of the positive and finite rank integral operator Q N,W , defined on L 2 (−1/2, 1/2), with kernel K N (x, y) = sin(N π(x−y)) sin(π(x−y)) 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 λ k,N (W ). In his pioneer work [17], 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 Q N,W . In particular, we show that each eigenvalue λ 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 λ 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.2010 Mathematics Subject Classification. Primary 42A38, 15B52. Secondary 60F10, 60B20.Key words and phrases. Band-limited sequences, eigenvalues and eigenfunctions, discrete prolate spheroidal wave functions and sequences, eigenvalues distribution and decay rate.

No abstract

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super-exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWFs). These set of special functions have been introduced in [16] and [7] and they are defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator decay at a super-exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As a first application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a periodic Sobolev space. Our second application is related to a generalization in the context of the GPSWFs, of the Landau-Pollak approximate dimension of the space of band-limited and almost time-limited functions, given in the context of the classical PSWFs, the eigenfunctions of the finite Fourier transform operator. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

No abstract