Mean convergence of prolate spheroidal series and their extensions

Mourad Boulsane (1) , Philippe Jaming (2) , Ahmed Souabni

1 University of Carthage
2 IMB - Institut de Mathématiques de Bordeaux

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).

Mots clés: mean convergence; prolate spheroidal wave function

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Variables complexes [math.CV]

Mourad Boulsane, Philippe Jaming, Ahmed Souabni. Mean convergence of prolate spheroidal series and their extensions. 2018. ⟨hal-01755912v2⟩