Fractal Sobolev systems of functions associated with orthonormal systems of functions

Abstract

This paper introduces the [Formula: see text]-fractal Sobolev system of functions corresponding to Sobolev orthonormal system of functions. An approximation-related result similar to Weierstrass theorem is derived. It is shown that the set of [Formula: see text]-fractal versions of Sobolev sums is dense and complete in the weighted Sobolev space [Formula: see text]. A Schauder basis and a Riesz basis of fractal type for the space [Formula: see text] are found. The Fourier–Sobolev expansion of an [Formula: see text]-fractal function [Formula: see text] corresponding to a certain set of interpolation points is presented. Moreover, some results on convergence of Fourier–Sobolev expansion of [Formula: see text] with respect to uniform norm and Sobolev norm are established.