A note on the norm convergence by Vilenkin–Fejér means

Abstract

The main aim of this paper is to nd necessary and su cient conditions for the convergence of Fejér means in terms of the modulus of continuity on the Hardy spaces when 0 < ≤ 1 2 .

On the convergence of Fejér means of Walsh-Fourier series in the space H p

“…The author [14] proved that if F ∈ H p (0 < p < 1/2) and ω Hp (1/2 n , F ) = o 1/2 n(1/p−2) , as n → ∞, then (3) σ n F − F p → 0, when n → ∞.…”

“…The author [14] also consider endpoint case p = 1/2 and proved that if F ∈ H 1/2 and (4) ω H 1/2 (1/2 n , F ) = o 1/n 2 , as n → ∞, then σ n F − F 1/2 → 0, when n → ∞.…”

Walsh-Marcinkiewicz means and Hardy spaces

“…In Section 3 we study the rate of the deviant behaviour of the -th Marcinkiewicz mean. Next, as an application, following ideas of [26] for one-dimensional Fejér means (see conditions (1) and (2)), we give a necessary and sufficient condition for the convergence of Walsh-Marcinkiewicz means in terms of modulus of continuity on H (G 2 ), Section 4. With the aid of some useful inequalities given in the proof of our main theorem, in Section 5 we prove a strong convergence theorem for Marcinkiewicz means.…”

Bounded operators on Martingale Hardy spaces

“…The reason of the divergence of S Mn+1 f is that when 0 < p < 1 the Fourier coefficients of f ∈ H p are not uniformly bounded (see Tephnadze [145]). On the other hand, there exists an absolute constant c p , depending only on p, such that…”

“…The proof of next Lemma can be found in Tephnadze [145]: Lemma 4.4 Let x ∈ I s \I s+1 , s = 0, ..., N − 1. Then…”