Biorthogonality in Approximation

Abstract

Numerical approximation can be carried out by ascent or descent methods — or in a more explicit way by expansion methods (like truncation, telescoping, pre-iteration). We use general biorthogonal systems (BOGS) to describe procedures of the latter type. This setting leads easily to useful results and provides good insight. The basic task is to improve or shorten a given expression by changing the coefficients. Thereby one employs information comprised in the elements and functionals of the BOGS, More specifically we consider expansions of Fourier (Chebyshev) type in the univariate and bivariate case.