Strong Approximation by Marcinkiewicz Means of two-dimensional Walsh-Kaczmarz-Fourier Series

TL;DR

This paper investigates the exponential uniform strong approximation of two-dimensional Walsh-Kaczmarz-Fourier series using Marcinkiewicz means, establishing optimal convergence results for continuous functions.

Contribution

It proves the exponential uniform strong summability of Marcinkiewicz means of Walsh-Kaczmarz-Fourier series and shows the result is optimal.

Findings

Uniform strong summability to the function $f$ exponentially in the power $1/2$

The result is proven to be the best possible

Extension of approximation theory in Walsh-Kaczmarz-Fourier analysis

Abstract

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh-Kaczmarz-Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh-Kaczmarz-Fourier series of the continuous function $f$ is uniformly strong summable to the function $f$ exponentially in the power $1/2$. Moreover, it is proved that this result is best possible.