For fixed real numbers c > 0, α > − 1 2 , the finite Hankel transform operator, denoted by H α c is given by the integral operator defined on L 2 (0, 1) with kernel K α (x, y) = √ cxyJ α (cxy).To the operator H α c , we associate a positive, self-adjoint compact integral operator Q α c = c H α c H α c . Note that the integral operators H α c and Q α c commute with a Sturm-Liouville differential operator D α c . In this paper, we first give some useful estimates and bounds of the eigenfunctions ϕThese estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator D α c . If (µ n,α (c)) n and λ n,α (c) = c |µ n,α (c)| 2 denote the infinite and countable sequence of the eigenvalues of the operators H (α) c and Q α c , arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer n ≥ 0, λ n,α (c) is decreasing with respect to the parameter α. As a consequence, we show that for α ≥ 1 2 , the λ n,α (c) and the µ n,α (c) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work. 2010 Mathematics Subject Classification. Primary 42C10, 65L70. Secondary 41A60, 65L15. Key words and phrases. Finite Hankel transform operator, Sturm-Liouville operator, eigenfunctions and eigenvalues, prolate spheroidal wave functions, approximation of Hankel band-limited functions.

The aim of this paper is to establish the range of p's for which the expansion of a function f ∈ L p in a generalized prolate spheroidal wave function (PSWFs) basis converges to f in L p . Two generalizations of PSWFs are considered here, the circular PSWFs introduced by D. Slepian and the weighted PSWFs introduced by Wang and Zhang. Both cases cover the classical PSWFs for which the corresponding results has been previously established by Barceló and Cordoba.To establish those results, we prove a general result that allows to extend mean convergence in a given basis (e.g. Jacobi polynomials or Bessel basis) to mean convergence in a second basis (here the generalized PSWFs).2010 Mathematics Subject Classification. 42C10;42C40 .

For fixed reals c > 0, a > 0 and α > − 1 2 , the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions are the eigenfunctions of the finite Hankel transform operator, denoted by H α c , which is the integral operator defined on L 2 (0, 1) with kernel H α c (x, y) = √ cxyJ α (cxy). Also, they are the eigenfunctions of the positive, selfadjoint compact integral operator Q α c = cH α c H α c. The CPSWFs play a central role in many applications such as the analysis of 2d-radial signals. Moreover, a renewed interest in the CPSWFs instead of Fourier-Bessel basis is expected to follow from the potential applications in Cryo-EM and that makes them attractive for steerable of principal component analysis(PCA). For this purpose, we give in this paper precise non-asymptotic estimates for the eigenvalues of Q α c , within the three main regions of the spectrum of Q α c. Moreover, we describe a series expansion of CPSWFs with respect to the generalized Laguerre functions basis of L 2 (0, ∞) defined by ψ a n,α (x) = √ 2a α+1 x α+1/2 e − (ax) 2 2L α n (a 2 x 2), whereL α n is the normalized Laguerre polynomial.

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.

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