The approximation of almost time- and band-limited functions by their expansion in some orthogonal polynomials bases

The aim of this paper is to investigate the quality of approximation of almost time- and almost band-limited functions by its expansion in two classical orthogonal polynomials bases: the Hermite basis and the ultraspherical polynomials bases (which include Legendre and Chebyshev bases as particular cases). As a corollary, this allows us to obtain the quality of approximation in the L2L2-Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of convergence of the Legendre series expansion of the prolate spheroidal wave functions.