Polynomial approximation of $L^2$-functions

Abstract

Let $\mu$ be a given probability measure supported by a compact subset $[ a, b]\subset \R$. Given a function $f$ element of $L^2\big([ a, b], d\mu\big)$,  it has been shown, under some integrability conditions, that a continuous version of $f$ can be pointwisely and uniformly approximated by a sequence of polynomial functions. More precisely by a partial-sum of orthogonal polynomials in $L^2\big([ a, b], d\mu\big)$. As an application, the obtained approximation Theorem has been used to set up a polynomial interpolation algorithm of $L^2$-functions. The derived interpolation algorithm has been implemented and compared to standard ones, such as the spline-cubic one.