Abstract
The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.
Highlights
The question of almost everywhere convergence is highly celebrated in the theory of Fourier series.It is quite well-known for both Walsh–Paley and trigonometric Fourier series, that the behavior of the logarihmic means ()log =1 is very nice
For each function which is integrable on the unit interval, the logarithmic means converge to almost everywhere
The aim of this paper is to show that the Walsh–Paley and the Walsh–Kaczmarz system are di erent in this point of view
Summary
The question of almost everywhere convergence is highly celebrated in the theory of Fourier series It is quite well-known for both Walsh–Paley and trigonometric Fourier series, that the behavior of the logarihmic means (). Riesz’s logarithmic means of a function with respect to the trigonometric system. Behind of many results in the trigonometric case there is the fact that the Riesz’s logarithmic kernel function is everywhere nonnegative. With a more di cult way the authors of this paper veri ed that Riesz’s logarithmic kernels with respect the Walsh–Paley system take only nonnegative values.
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