George Tephnadze

Fejér means of Vilenkin-Fourier series

The main aim of this paper is to prove that there exist a martingale f ∈ H1/2 such that Fejer means of Vilenkin-Fourier series of the martingale f is not uniformly bounded in the space L1/2.

On the partial sums of Vilenkin-Fourier series

The aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. Also, the obtained results we use to prove approximation and strong convergence theorems on the martingale Hardy spaces Hp, when 0 < p ≤ 1.

On the maximal operators of Walsh-Kaczmarz-Fejér means

AbstractThe main aim of this paper is to prove that the maximal operator

Strong Convergence of Two-Dimensional Walsh–Fourier Series

\sigma _p^{\kappa , * } f: = \sup _{n \in P} {{\left| {\sigma _n^\kappa f} \right|} \mathord{\left/ {\vphantom {{\left| {\sigma _n^\kappa f} \right|} {\left( {n + 1} \right)^{{1 \mathord{\left/ {\vphantom {1 {p - 2}}} \right. \kern-\nulldelimiterspace} {p - 2}}} }}} \right. \kern-\nulldelimiterspace} {\left( {n + 1} \right)^{{1 \mathord{\left/ {\vphantom {1 {p - 2}}} \right. \kern-\nulldelimiterspace} {p - 2}}} }}

On the Vilenkin–Fourier coefficients

is bounded from the Hardy space Hp to the space Lp for 0 < p < 1/2.

Strong convergence of two-dimensional Vilenkin-Fourier series

Abstract The main aim of this paper is to find necessary and sufficient conditions for the convergence of Fejér means in terms of the modulus of continuity on the Hardy spaces Hp when 0 < p ≤ 1/2.

Extension of the unit normal vector field from a hypersurface

We prove that certain means of quadratic partial sums of the two-dimensional Walsh–Fourier series are uniformly bounded operators acting from the Hardy space Hp to the space Lp for 0 < p < 1.

Walsh-Marcinkiewicz means and Hardy spaces

In [14] we investigated some Vilenkin—Norlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H1/(1+α) to the space weak-L1/(1+α), (0 < α ≦ 1). In this paper we construct a martingale in the space H1/(1+α), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin—Norlund means with non-increasing coefficients are not bounded from the Hardy space H1/(1+α) to the space L1/(1+α). In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.