  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{\'o}
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).
