ON THE NEGATIVITY OF THE WALSH–KACZMARZ–RIESZ LOGARITHMIC KERNELS

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.


INTRODUCTION
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 1 log =1 ( ) is very nice. That is, for each function which is integrable on the unit interval, the logarithmic means converge to almost everywhere.
In [5] Hardy gave a necessary and su cient condition for the convergence of a certain point of Riesz's logarithmic means of a function with respect to the trigonometric system. For some material concerning the trigonometric system see the book of Zygmund [2]. With respect to Walsh-Paley-Riesz logarithmic means see e.g. the paper of Gát and Goginava [4]. Behind of many results in the trigonometric case there is the fact that the Riesz's logarithmic kernel function is everywhere nonnegative. This very useful property of the kernel is given with the help of Abel transform and the fact that the Fejér kernels are everywhere nonnegative. On the other hand, it is quite simple to give Walsh-Paley-Fejér kernels which take negative values. That is, the method used in the trigonometric case does not work as a proof of the nonnegativity. However, 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. This result will be published elsewhere. 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. More precisely, we give a sequence of Walsh-Kaczmarz-Riesz logarithmic kernels such that all of them take negative values on some intervals. Next, we give some necessary preliminaries.
By means of the transformation ∶ → , ( ∈ ℕ) which is clearly measure-preserving and such that ( ( )) = we will be able to discuss the values of Walsh-Kaczmarz functions and kernels. That is, we have (see also [6]) the following important equality: It is known [1] that the system ( , ∈ ℕ) is the character system of ( , +), where the group operation + is the so-called dyadic or logical addition on . That is, for any , ∈ Let be either or . Denote by Unauthenticated | Downloaded 11/05/21 10:51 PM UTC the Dirichlet, the Fejér or ( , 1) kernels and Walsh-logarithmic means, respectively. If in the notation is missing, then we mean the Walsh-Paley ordering. Moreover: It is also known that the Fourier coe cient, the th partial sum of Fourier series and the Fejér or

The key observation
In the sequel we will need the following lemma. The main aim of this paper is to prove the following negativity result concerning the Riesz's logarithmic kernels with respect to the Walsh-Kaczmarz system.

PROOFS
Proof of Lemma 2.1. Use the formula of Skvortsov for Walsh-Kaczmarz-Dirichlet functions in [6].