Matrix Summability of Walsh-Fourier Series

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 Cesaro means with varying parameters.