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

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 $f\in L^1([0,1{)}^2)$, $f\ge 0$ one has for almost every $z\in [0,1{)}^2$ that $$\underset{n,m\to \infty ;n,m\in \mathcal{N}}{\lim }{S}_{n,m}f(z)=f(z)\text{ and }\underset{n,m\to \infty ;n,m\in \mathcal{M}}{\mathrm{limsup}}|S_{n,m}f(z)|=\infty .$$ 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.