On the partial sums of Vilenkin-Fourier series

Abstract

The main aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. We also use our results to prove approximation and strong convergence theorems on the martingale Hardy spaces Hp, when 0 < p ≤ 1.2010 Mathematics Subject Classification. 42C10.

Strong convergence theorem for Vilenkin--Fejer means

“…. Since (see (5) and Lemma 2) (22) K Mn (x) = D Mn (x) = 0, for n ≥ 2 from (15) and (20)- (21) we obtain that…”

“…where S k f denotes the k-th partial sum of the Vilenkin-Fourier series of f. (For the trigonometric analogue see in Smith [17], for the Walsh-Paley system in Simon [15]). Simon [16] (see also [23]) proved that there exists an absolute constant c p , depending only on p, such that for all f ∈ H p and n ∈ P + , where [p] denotes integer part of p. In [21] it was proved that sequence 1/k 2−p ∞ k=1 (0 < p < 1) in (1) are given exactly. Weisz [27] considered the norm convergence of Fejér means of Walsh-Fourier series and proved the following:…”

A sharp boundedness result for restricted maximal operators of Vilenkin–Fourier series on martingale Hardy spaces

“…We also mention the following well-known consequences (for details see e.g. the books [18,19] and [14]):…”

On the partial sums of Walsh–Fourier series

“…The proof of estimate(20) is analogously to the proof of estimate (48). This completes the proof of theorem 3.…”