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The aim of this paper is to investigate the quality of approximation ofalmost time and almost band-limited functions by its expansion in threeclassical orthogonal polynomials bases: the Hermite, Legendre and Chebyshevbases. As a corollary, this allows us to obtain the quality of approximation inthe L 2 --Sobolev space by these orthogonal polynomials bases. Also, we obtainthe rate of the Legendre series expansion of the prolate spheroidal wavefunctions. Some numerical examples are given to illustrate the differentresults of this work.

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