arXiv subjects

Ahmed Souabni

Publications and source records attributed to Ahmed Souabni.

7 recordsLinked to original sources

arXiv2025

Expansion into Clifford Prolate Spheroidal Wave Functions

In this paper, we investigate the properties of Clifford prolate spheroidal wave functions (CPSWFs) through their associated eigenvalues. We prove that the expansion coefficients in CPSWFs series decay as both the order and the homogeneity degree increase. By establishing a precise connection between the radial CPSWFs and the eigenfunctions of the finite Hankel transform, we derive explicit and non-asymptotic bounds on the corresponding eigenvalues and transfer the spectral decay estimates to the Clifford setting. Consequently, we obtain super-exponential decay rates for the CPSWF expansion coefficients of band-limited Clifford-valued functions. Numerical experiments illustrate both the accuracy and the efficiency of these approximations.

math.GM↗

arXiv2025

Eigenvalue distribution of some random matrices

In this paper, we investigate the eigenvalue distribution of a class of kernel random matrices whose $(i,j)$-th entry is $f(X_i,X_j)$ where $f$ is a symmetric function belonging to the Paley-Wiener space $\mathcal{B}_c$ and $(X_i)_{1\leq i \leq N}$ are i.i.d. random variables. We rigorously prove that, with high probability, the eigenvalues of these random matrices are well approximated by those of an underlying estimator. A particularly notable case is when $f=sinc$ , which has been widely studied due to its relevance in various scientific fields, including machine learning and telecommunications. In this case, we push forward the general approach by computing the eigenvalues of the estimator. More precisely, we have proved that the eigenvalues are concentrated around zero and one. In particular, we address the case of large values of $c$ with respect to the matrix size $N$, which, to the best of our knowledge, has not been studied in the literature. Furthermore, we establish that the frequency of eigenvalues close to one is proportional to $c$. Numerical results are provided in order to illustrate the theoretical findings.

math.ST↗

arXiv2023

Further properties of ball prolates and approximation of related almost band-limited functions

In this paper we aim to give various explicit and local estimates of ball prolate spheroidal wave functions defined in [25] as eigenfunctions of both finite Fourier transform and some differential operator. In particular, we give further refined bounds of these functions and their related eigenvalues. As consequence, we show that ball PSWFs are well adapted for the approximation of almost band-limited functions and we compare this result with the one related to the ball polynomials.

math.CA↗

arXiv2019

Further Spectral Properties of the Weighted Finite Transform Operator and Approximation in Weighted Sobolev Spaces

In this work, we first give some mathematical preliminairies concerning the generelized prolate spheroidal wave functions(GPSWFs). This set of special functions have been introduced in [21]and [13] 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 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.

math.CA↗

arXiv2018

Mean convergence of prolate spheroidal series and their extensions

The aim of this paper is to establish the range of p's for which the expansion of a function f $\in$ 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 Barcel{ó} 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).

math.CA↗

arXiv2015

Weighted finite Fourier transform operator: Uniform approximations of the eigenfunctions, eigenvalues decay and behaviour

In this paper, we first give two uniform asymptotic approximations of the eigenfunctions of the weighted finite Fourier transform operator, defined by ${\displaystyle \mathcal F_c^{(α)} f(x)=\int_{-1}^1 e^{icxy} f(y)\,(1-y^2)^α\, dy,\,}$ where $ c >0, α> -1$ are two fixed real numbers. The first uniform approximation is given in terms of a Bessel function, whereas the second one is given in terms of a normalized Jacobi polynomial. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs). By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator $\mathcal F_c^{(α)}$ in the case where $0<α< 3/2.$ Finally, by computing the trace and an estimate of the norm of the operator ${\displaystyle \mathcal Q_c^α=\frac{c}{2π} \mathcal F_c^{α^*} \mathcal F_c^α,}$ we give a lower and an upper bound for the counting number of the eigenvalues of $Q_c^α,$ when $c>>1.$

math.CA↗

arXiv2015

Generalized Prolate Spheroidal Wave Functions: Spectral Analysis and Approximation of Almost Band-limited Functions

In this work, we first give various explicit and local estimates of the eigenfunctions of a perturbed Jacobi differential operator. These eigenfunctions generalize the famous classical prolate spheroidal wave functions (PSWFs), founded in 1960's by D. Slepian and his co-authors and corresponding to the case $α=β=0.$ They also generalize the new PSWFs introduced and studied recently in \cite{Wang2}, denoted by GPSWFs and corresponding to the case $α=β> -1.$ The main content of this work is devoted to the previous interesting special case $α=β.$ In particular, we give further computational improvements, as well as some useful explicit and local estimates of the GPSWFs. More importantly, by using the concept of a restricted Paley-Wiener space, we relate the GPSWFs to the solutions of a generalized energy maximisation problem. As a consequence, many desirable spectral properties of the self-adjoint compact integral operator associated with the GPSWFs are deduced from the rich literature of the PSWFs. In particular, we show that the GPSWFs are well adapted for the spectral approximation of the classical $c-$band-limited as well as almost $c-$band-limited functions. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

math.CA↗