## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 9, Number 1 (2018), 137-150.

### On an approximation of $2$-dimensional Walsh–Fourier series in martingale Hardy spaces

L. E. Persson, G. Tephnadze, and P. Wall

#### Abstract

In this paper, we investigate convergence and divergence of partial sums with respect to the $2$-dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh–Fourier series. We also show that these conditions are in a sense necessary and sufficient.

#### Article information

**Source**

Ann. Funct. Anal., Volume 9, Number 1 (2018), 137-150.

**Dates**

Received: 29 November 2016

Accepted: 11 March 2017

First available in Project Euclid: 5 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1507169076

**Digital Object Identifier**

doi:10.1215/20088752-2017-0032

**Mathematical Reviews number (MathSciNet)**

MR3758749

**Zentralblatt MATH identifier**

1382.42017

**Subjects**

Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Secondary: 42B25: Maximal functions, Littlewood-Paley theory

**Keywords**

Fourier series Walsh system strong summability martingale Hardy space 2-dimensional modulus of continuity

#### Citation

Persson, L. E.; Tephnadze, G.; Wall, P. On an approximation of $2$ -dimensional Walsh–Fourier series in martingale Hardy spaces. Ann. Funct. Anal. 9 (2018), no. 1, 137--150. doi:10.1215/20088752-2017-0032. https://projecteuclid.org/euclid.afa/1507169076