Strong Summability of Two-Dimensional Vilenkin Fourier series

TL;DR

This paper investigates the exponential uniform strong summability of two-dimensional Vilenkin-Fourier series, demonstrating that such series of continuous functions converge uniformly and optimally at an exponential rate with power 1/2.

Contribution

It establishes the exponential uniform strong summability of two-dimensional Vilenkin-Fourier series and proves the optimality of this convergence rate.

Findings

Series are uniformly strongly summable to the function exponentially in the power 1/2

The result is proven to be the best possible

Convergence holds for continuous functions

Abstract

In this paper we study the exponential uniform strong summability of two-dimensional Vilenkin-Fourier series. In particular, it is proved that the two-dimensional Vilenkin-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.