A Generalization for Fourier Transforms of a Theorem due to Marcinkiewicz

Abstract

It is shown that the maximal operator of the Marcinkiewicz means of a temperedŽ . quently, is of weak type 1, 1 , where p -1. As a consequence we obtain a 0 generalization for Fourier transforms of a summability result due to Marcinkiewicz Ž 2 . and Zhizhiashvili, more exactly, the Marcinkiewicz means of a function f g L R 1 converge a.e. to the function in question. Moreover, we prove that the Ž 2 . Marcinkiewicz means are uniformly bounded on the spaces H R and so they

Almost Everywhere Convergence of Fejér Means of Two-dimensional Triangular Walsh–Fourier Series

“…. , (19) different from zero. If say, (n + k) a = 0 for some t 1 + i + m < a < l, then as k t 1 +i+m changes from 0 to 1, we do not have change in (n + k) l and in ω (n+k) (t 2 +1) (x 2 ) and consequently (19) is zero.…”

“…, (n + k) t 2 should be 1 for each k i (0 ≤ i ≤ t 2 , i = t 1 + i + m, l + j) for those addends in (19) different from zero. If say, (n + k) a = 0 for some t 1 + i + m < a < l, then as k t 1 +i+m changes from 0 to 1, we do not have change in (n + k) l and in ω (n+k) (t 2 +1) (x 2 ) and consequently (19) is zero. That is, the number is the tuples (k 0 , .…”

“…We have a fix n and if k t 1 +i+m , k l+j runs in {0, 1} with fixed other indices of k, then to avoid (19) to be zero all the coordinates of (n + k) t 1 +i+m+1 , . .…”

“…relation t n f → f with respect to the trigonometric system. The "L 1 result" for the trigonometric and the Walsh-Paley system see the papers of Zhizhiasvili [22] (trigonometric system), Weisz [19] (Walsh system) and Goginava [9,8] (Walsh system). Some of these results (including the proofs) can also be found in [20].…”

Marcinkiewicz-like means of two dimensional Vilenkin--Fourier series

“…Dyachenko [3] proved this result for dimensions greater than 2. In 2001, Weisz [12] proved this result with respect to the Walsh-Paley system. Goginava [6] proved the corresponding result for the d-dimensional Walsh-Paley system.…”