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Divergent Fourier Series with Respect to Biorthonormal Systems in Function Spaces Near $L^1$

Nikoloz Devdariani

TL;DR

The paper addresses the problem of extending Bochkarev-type divergence results for Fourier series, with respect to uniformly bounded biorthonormal systems, to variable exponent Lebesgue spaces near $L^1$ in several variables. It develops a framework identifying the class of exponent functions $p(\cdot)$, via the set $P_{\ln}$ and rearrangement $p^*(t)$, for which a Bochkarev-type divergence holds in $L^{p(\omega(\cdot))}(\Omega)$ after a measure-preserving transformation $\omega$, yielding divergence on a set of positive measure for appropriate $F_1,F_2\in L^{p(\omega(\cdot))}$. The work establishes necessary and sufficient conditions for the closedness of $C(\Omega)$ in variable-exponent spaces and extends the Bochkarev construction to the multi-variable setting, bridging classical $L^1$ divergence phenomena with nonstandard growth spaces. The results provide a rigorous characterization of when Fourier series with respect to biorthonormal systems can diverge on positive-measure sets within near-$L^1$ spaces, under a systematic extension to higher dimensions and variable exponent growth.

Abstract

In this paper, we generalize Bochkarev's theorem, which states that for any uniformly bounded biorthonormal system $Φ$, there exists a Lebesgue integrable function whose Fourier series with respect to the system $Φ$ diverges on a set of positive measure. We find the class of variable exponent Lebesgue spaces $L^{p(\cdot)}([0,1]^n)$, where $1 < p(x) < \infty$ almost everywhere on $[0,1]^n$, such that the aforementioned Bochkarev's theorem holds.

Divergent Fourier Series with Respect to Biorthonormal Systems in Function Spaces Near $L^1$

TL;DR

The paper addresses the problem of extending Bochkarev-type divergence results for Fourier series, with respect to uniformly bounded biorthonormal systems, to variable exponent Lebesgue spaces near in several variables. It develops a framework identifying the class of exponent functions , via the set and rearrangement , for which a Bochkarev-type divergence holds in after a measure-preserving transformation , yielding divergence on a set of positive measure for appropriate . The work establishes necessary and sufficient conditions for the closedness of in variable-exponent spaces and extends the Bochkarev construction to the multi-variable setting, bridging classical divergence phenomena with nonstandard growth spaces. The results provide a rigorous characterization of when Fourier series with respect to biorthonormal systems can diverge on positive-measure sets within near- spaces, under a systematic extension to higher dimensions and variable exponent growth.

Abstract

In this paper, we generalize Bochkarev's theorem, which states that for any uniformly bounded biorthonormal system , there exists a Lebesgue integrable function whose Fourier series with respect to the system diverges on a set of positive measure. We find the class of variable exponent Lebesgue spaces , where almost everywhere on , such that the aforementioned Bochkarev's theorem holds.
Paper Structure (3 sections, 6 theorems, 64 equations)

This paper contains 3 sections, 6 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. definitions and auxiliary results
  3. Proof of Theorem \ref{['thm_main']}

Key Result

Theorem 1.2

Let $X$ be a separable metric space with a Borel regular outer measure $\mu^*$ such that $\mu^*(X) = 1$. Then, for any biorthonormal system $\{f_n, g_n\}$ satisfying conditions and for any measurable set $E \subset X$. Then there exist functions $F_1, F_2\in L^1(X, \mu)$ and a set $E\subset X$ such that and for all $x\in E$ we have

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2: Edmunds, Lang, Nekvinda
  • Theorem 2.3
  • Theorem 2.4