Uniform and L–convergence of Logarithmic Means of Walsh–Fourier Series

Abstract

The (Norlund) logarithmic means of the Fourier series of the integrable function f is: $$ \begin{array}{*{20}c} {{\frac{1} {{l_{n} }}{\sum\limits_{k = 1}^{n - 1} {\frac{{S_{k} {\left( f \right)}}} {{n - k}}} },}} & {{{\text{where}}\;l_{n} : = {\sum\limits_{k = 1}^{n - 1} {\frac{1} {k}} }.}} \\ \end{array} $$ In this paper we discuss some convergence and divergence properties of this logarithmic means of the Walsh–Fourier series of functions in the uniform, and in the L1 Lebesgue norm. Among others, as an application of our divergence results we give a negative answer to a question of Moricz concerning the convergence of logarithmic means in norm.