On the a.e. convergence of Walsh-Kaczmarz-Fourier series

Abstract

Abstract.It is shown that partial sums of Walsh-KaczmarzFourier series of functions in the Orlicz class L(log+ L)2 converge a.e. The proof utilizes an estimate of P. Sjölin on the partial sums of the usual Walsh-Fourier series.

On the maximal operator of Walsh-Kaczmarz-Fejér means

“…In 2008 Goginava [8] proved that the maximal operatorσ * In 1948 Šneider [16] introduced the Walsh-Kaczmarz system and showed that the inequality lim sup n→∞ D κ n (x) log n C > 0 holds a.e. In 1974 Schipp [12] and Young [18] proved that the Walsh-Kaczmarz system is a convergence system. Skvortsov in 1981 [15] showed that the Fejér means with respect to the Walsh-Kaczmarz system converge uniformly to f for any continuous function f .…”

On the maximal operator of (C, α)-means of Walsh–Kaczmarz–Fourier series

“…In 1974, Schipp [2] and Young [3] proved that the Walsh-Kaczmarz system is a convergence system. In 1981, Skvortsov [4] showed that the Fejér means with respect to the Walsh-Kaczmarz system converge uniformly to f for any continuous function f. Gát [5] proved that, for any integrable function, the Fejér means with respect to the Walsh-Kaczmarz system converge almost everywhere to this function and, furthermore,…”

Оценки Ядра Дирихле И Расходящиеся Ряды Фурье По Системе Уолша - Качмажа

“…В работе [2] показано, что для всякой функции из класса (ln + ) 2 [0, 1] ее ряды Фурье-Уолша-Качмажа сходятся к ней почти всюду. Для тригонометрической системы и систе-мы Уолша-Пэли аналогичный результат для класса ln + ln + ln + ln + [0, 1] полу-чен Антоновым в [3] и [4], а также независимо Сьелином и Сориа для системы Уолша-Пэли в [5].…”