Matrix Summability of Walsh–Fourier Series

Ushangi Goginava and Károly Nagy
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Ushangi Goginava: Department of Mathematical Sciences, United Arab Emirates University, Al Ain P.O. Box 15551, United Arab Emirates
Károly Nagy: Institute of Mathematics and Computer Sciences, Eszterházy Károly Catholic University, Leányka Street 4, H3300 Eger, Hungary

Mathematics, 2022, vol. 10, issue 14, 1-25

Abstract: The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in L 1 space and in C W space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator t ∗ ( f ) of the matrix transform of the Walsh–Fourier series. As a consequence, we obtain the sufficient condition so that the matrix transforms t n ( f ) of the Walsh–Fourier series are convergent almost everywhere to the function f . The problems listed above are related to the corresponding Lebesgue constant of the matrix transformations. The paper sets out two-sides estimates for Lebesgue constants. The proven theorems can be used in the case of a variety of summability methods. Specifically, the proven theorems are used in the case of Cesàro means with varying parameters.

Keywords: Walsh system; matrix transforms; Cesaro mean; logarithmic means; martingale transform; weak type inequality; convergence in norm; almost everywhere convergence and divergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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