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

TL;DR

This paper extends Bochkarev's theorem to variable exponent Lebesgue spaces, showing divergence of Fourier series with respect to biorthonormal systems on sets of positive measure.

Contribution

It identifies conditions on variable exponent Lebesgue spaces where Bochkarev's divergence theorem applies, generalizing previous results.

Findings

Divergence of Fourier series in variable exponent spaces

Characterization of spaces where divergence occurs

Extension of Bochkarev's theorem to new function spaces

Abstract

In this paper, we generalize Bochkarev's theorem, which states that for any uniformly bounded biorthonormal system $\Phi$, there exists a Lebesgue integrable function whose Fourier series with respect to the system $\Phi$ 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.