On the coexistence of divergence and convergence phenomena for the Fourier-Haar series for non-negative functions

TL;DR

This paper investigates the simultaneous divergence and convergence behaviors of Fourier-Haar series for non-negative functions, providing necessary and sufficient conditions for such phenomena using low-discrepancy sequences.

Contribution

It establishes a precise criterion on index sets for the coexistence of divergence and convergence in Fourier-Haar series for non-negative functions.

Findings

Characterizes conditions for convergence of Fourier-Haar partial sums.

Identifies conditions for divergence of Fourier-Haar partial sums.

Extends previous results using low-discrepancy sequences.

Abstract

Let $\{H_{n,m}\}_{n,m\in \mathbb{N}}$ be the two dimensional Haar system and $S_{n,m}f$ be the rectangular partial sums of its Fourier series with respect to some $f\in L^1([0,1)^2)$. Let $\mathcal{N}, \mathcal{M}\subset \mathbb{N}$ be two disjoint subsets of indices. We give a necessary and sufficient condition on the sets $\mathcal{N}, \mathcal{M}$ so that for some $f \in L^1([0,1)^2)$, $f \geq 0$ one has for almost every $z\in [0,1)^2$ that $$ \lim_{n,m \rightarrow \infty;n,m \in \mathcal{N}}S_{n,m}f(z)=f(z)\quad \text{ and }\quad \limsup_{n,m \rightarrow \infty;n,m \in \mathcal{M}}|S_{n,m}f(z)|=\infty. $$ The proof uses some constructions from the theory of low-discrepancy sequences such as the van der Corput sequence and an associated tiling of the plane. This extends some earlier results.