The partial sum process of orthogonal expansion as geometric rough process with Fourier series as an example---an improvement of Menshov-Rademacher theorem

TL;DR

This paper demonstrates that the partial sum process of orthogonal series, including Fourier series, is a geometric 2-rough process under certain conditions, improving the Menshov-Rademacher theorem with a refined criterion for Fourier series.

Contribution

It extends the Menshov-Rademacher theorem by showing the partial sum process is a geometric 2-rough process and refines the condition for Fourier series.

Findings

Partial sum process of orthogonal series is a geometric 2-rough process.

Condition for Fourier series can be improved.

Identifies an equivalent condition on the limit function.

Abstract

In this paper, we prove that the partial sum process of general orthogonal series is a geometric 2-rough process under the same condition as in Menshov-Rademacher Theorem. For Fourier series, the condition can be improved, and an equivalent condition on the limit function is identified.