Approximation in M\"untz spaces $M_{\Lambda ,p}$ of $L_p$ functions for $1<p<\infty $ and bases

TL;DR

This paper investigates approximation properties and basis existence in M"untz spaces of $L_p$ functions, demonstrating their relation to Weil-Nagy's class and establishing conditions for Schauder bases.

Contribution

It shows that M"untz spaces satisfying certain conditions are contained in Weil-Nagy's class and explores the existence of Schauder bases within these spaces.

Findings

M"untz spaces are contained in Weil-Nagy's class after isomorphism and variable change.

Existence of Schauder bases in M"untz spaces is established.

Fourier approximation methods are effectively applied to M"untz spaces.

Abstract

M\"untz spaces satisfying the M\"untz and gap conditions are considered. A Fourier approximation of functions in the M\"untz spaces $M_{\Lambda ,p}$ of $L_p$ functions is studied, where $1<p<\infty $. It is proved that up to an isomorphism and a change of variables these spaces are contained in Weil-Nagy's class. Moreover, existence of Schauder bases in the M\"untz spaces $M_{\Lambda ,p}$ is investigated.