Construction of schauder decomposition on banach spaces of periodic functions

Abstract

This paper deals with Schauder decompositions of Banach spaces X2x of 2π-periodic functions by projection operators Pk onto the subspaces Vk, k = 0,1,..., which form a multiresolution of X2x. The results unify the study of wavelet decompositions by orthogonal projections in the Hilbert space L2x 2 on one hand and by interpolatory projections in the Banach space C2x on the other. The approach, using "orthogonal splines", is constructive and leads to the construction of a Schauder decomposition of X2x and a biorthogonal system for X2x and its dual X2x*. Decomposition and reconstruction algorithms are derived from the construction.