Pointwise Convergence of Dyadic Partial Sums of Almost Periodic Fourier Series

TL;DR

This paper proves pointwise convergence of dyadic partial sums of almost periodic Fourier series in Stepanov spaces, extending classical results and establishing new bounds for related operators.

Contribution

It extends convergence results to Stepanov spaces $S^p$ for all $p ext{ in } (1, \infty)$ and develops general bounds for operators like the Hilbert transform.

Findings

Boundedness of maximal dyadic partial sum operator on $S^{2^k}$ spaces.

Establishment of a Littlewood--Paley type theorem in Stepanov spaces.

Potential extension of results to all $S^p$ spaces based on a conjecture.

Abstract

It is a classical result that dyadic partial sums of the Fourier series of functions $f \in L^p(\mathbb{T})$ converge almost everywhere for $p \in (1, \infty)$. In 1968, E. A. Bredihina established an analogous result for functions belonging to the Stepanov space of almost periodic functions $S^2$ whose Fourier exponents satisfy a natural separation condition. Here, the maximal operator corresponding to dyadic partial summation of almost periodic Fourier series is bounded on the Stepanov spaces $S^{2^k}$, $k \in \mathbb{N}$ for functions satisfying the same condition; Bredihina's result follows as a consequence. In the process of establishing these bounds, some general results are obtained which will facilitate further work on operator bounds and convergence issues in Stepanov spaces. These include a boundedness theorem for the Hilbert transform and a theorem of Littlewood--Paley type. An improvement of "$S^{2^k}$, $k \in \mathbb{N}$" to "$S^p$, $p \in (1, \infty)$" is also seen to follow from a natural conjecture on the boundedness of the Hilbert transform.