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

Abstract

The main aim of this paper is to find the necessary and sufficient conditions for a modulus of continuity of a martingale F ∈ Hp, for which Fejér means convergence in Hp-norm, when 0 < p ≤ 1/2. 2010 Mathematics Subject Classification. 42C10.

Some inequalities related to strong convergence of Riesz logarithmic means

“…Remark 3. Similar problem for partial sums and Fejer means with respect to Walsh and Vilenkin systems can be found in Tephnadze [39], [40] and [41].…”

“…Remark 4. Approximation properties of some summability methods in the classical and real Hardy spaces were considered by Oswald [32], Kryakin and Trebels [18], Storoienko [30], [31] and for martingale Hardy spaces in Fridli, Manchanda and Siddiqi [9] (see also [7], [8]), Nagy [20], [21], [22], Tephnadze [39], [40], [41].…”

The One-dimensional Martingale Hardy Spaces and Partial Sums and Fejér Means with respect to Walsh system

“…In the third chapter of this thesis (see also [62]) we find rate of divergence of subsequences of Fejér means of the one-dimensional Walsh-Fourier series on the martingale Hardy spaces…”

“…In the third chapter of this thesis we find necessary and sufficient conditions for the modulus of continuity, for which subsequences σ n k f of Fejér means of the one-dimensional Walsh-Fourier series converge in H p (G) norm (see also [62]).…”

Bounded operators on Martingale Hardy spaces

“…, Explicit constructions of H p martingales can be found in the papers [5], [104], [110], [162], [159], [157], [158], [152], [160], [155], [161], [113].…”