Triangular Cesàro summability of two dimensional Fourier series

Abstract

It is proved that the maximal operator of the triangular Cesàro means of a two-dimensional Fourier series is bounded from the periodic Hardy space Hp(T 2 ) to Lp(T 2 ) for all 2/(2 + α) < p ∞ and, consequently, is of weak type (1, 1). As a consequence we obtain that the triangular Cesàro means of a function f ∈ L1(T 2 ) converge a.e. to f .

Maximal operator of the Fejér means of triangular partial sums of two-dimensional Walsh–Fourier series

“…This summability method is rarely investigated in the literature (see the references in [15]) and has not been studied yet for Walsh-Fourier series. The main aim of this paper is to consider the triangular summability method for two-dimensional Walsh-Fourier series and to generalize the results just mentioned.…”

Фейеровская Суммируемость По Треугольникам Двойных Рядов Фурье

“…Integrating over I N × I N . Since a ∞ ≤ c 2N/p from Lemma 7 we obtain (23) σ △ n a (x, y) ≤ a ∞ I N ×I N K △ n (x + s, y + t) dµ (s, t)…”

“…of the two-dimensional trigonometric Fourier series. This summability method is rarely investigated in the literature (see the references in [23]). In [12] it is proved that the maximal operator σ △ # := sup n σ △ 2 n f of the Fejér means of the triangular partial sums of the double Walsh-Fourier series is bounded from the dyadic Hardy space…”

“…S △ j f of the two-dimensional trigonometric Fourier series. This summability method is rarely investigated in the literature (see the references in [23]). In [12] it is proved that the maximal operator σ △ # := sup n σ △ 2 n f of the Fejér means of the triangular partial sums of the double Walsh-Fourier series is bounded from the dyadic Hardy space H p (G × G) to the L p (G × G) if p > 1/2, is bounded from H 1/2 (G × G) to the space weak-L 1/2 (G × G) and it is not bounded from H 1/2 (G × G) to L 1/2 (G × G).…”

Summability of Multi-Dimensional Trigonometric Fourier Series

“…. , d. We proved in [102,101] that Theorems 6.1, 6.2, 8.1, 9. Here, we give only some hints for the proofs.…”