An Upper Bound for the Menchov-Rademacher Operator for Right Triangles

TL;DR

This paper extends the Menchov-Rademacher inequality to two-parameter orthogonal series generated by right triangles, establishing almost everywhere convergence under bounded eccentricity.

Contribution

It introduces an analogue of the Menchov-Rademacher inequality specifically for right triangles, enabling convergence analysis for two-parameter orthogonal series.

Findings

Derived an inequality for right triangle series

Proved almost everywhere convergence with bounded eccentricity

Extended harmonic analysis tools to geometric series

Abstract

The Menchov-Rademacher inequality is an inequality in harmonic analysis that bounds the $L_2$ norm of a certain maximal operator. It was first established in order to prove almost everywhere convergence of a one-parameter series of orthogonal functions. When two-parameter series of orthogonal functions is considered, the exact way the series is grouped becomes essential. We will consider grouping of a two-parameter series, generated by a sequence of right triangles with a vertex at the origin, who might be non-equilateral, and prove almost everywhere convergence when the eccentricity of those triangles is bounded. In order to carry out the proof, we will derive an analogue of the Menchov-Rademacher inequality for right triangles.