Vilenkin-Fourier series in variable Lebesgue spaces

TL;DR

This paper characterizes the conditions on variable Lebesgue space exponents under which Vilenkin-Fourier series partial sums converge to the original function.

Contribution

It provides a complete characterization of exponent functions ensuring convergence of Vilenkin-Fourier series in variable Lebesgue spaces.

Findings

Identifies all variable exponents for convergence

Establishes convergence criteria in variable Lebesgue spaces

Extends classical Fourier analysis results

Abstract

Let $S_{n}f$ denote the $n$th partial sum of the Vilenkin-Fourier series of a function $f \in L^{1}(G)$. For $1 < p_{-} \leq p_{+} < \infty$, we characterize all exponents $p(\cdot)$ for which the convergence of $S_{n}f$ to $f$ in $L^{p(\cdot)}(G)$ holds whenever $f \in L^{p(\cdot)}(G)$.