Menshov type correction theorems for sequences of compact operators

TL;DR

This paper extends Menshov type correction theorems to sequences of compact operators, enhancing understanding of Fourier series convergence in trigonometric and Walsh systems through weak-$L^1$ estimates.

Contribution

It introduces Menshov type correction theorems for compact operator sequences, clarifying key weak-$L^1$ estimates crucial for Fourier series correction results.

Findings

Established Menshov correction theorems for compact operators.

Reproduced several classical Fourier series correction results.

Highlighted the importance of weak-$L^1$ estimates in this context.

Abstract

We prove Menshov type "correction" theorems for sequences of compact operators, recovering several results of Fourier series in trigonometric and Walsh systems. The paper clarifies the main ingredient, which is important in the study of such "correction" theorems. That is the weak-$L^1$ estimate for the maximal Fourier sums of indicator functions of some specific sets.