On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces

On The maximal operators of Vilenkin-Fejér means

The main aim of this paper is to prove that the maximal operator $\overset{% \sim }{σ}^{*}f:=\underset{n\in P}{\sup }\frac{\left| σ_{n}f\right| }{\log ^{2}\left( n+1\right) }$ is bounded from the Hardy space $H_{1/2}$ to the space $L_{1/2}$, where $σ_{n}f$ is Fejér means of bounded Vilenkin-Fourier series.

On The maximal operators of Vilenkin-Fejér means on Hardy spaces

The main aim of this paper is to prove that when $0<p<1/2$ the maximal operator $\overset{\sim }{σ}_{p}^{\ast }f:=\underset{n\in \mathbb{N}}{% \sup }\frac{\left\vert σ_{n}f\right\vert }{\left( n+1\right) ^{1/p-2}}$ is bounded from the martingale Hardy space $H_{p}$ to the space $L_{p},$ where $σ_{n}$ is $n$-th Fejér mean with respect to bounded Vilenkin system.

关于零测集上Vilenkin系统的Féjer平均的发散性

On the divergence of Féjer means with respect to Vilenkin systems on the set of measure zero

该研究针对零测集上Vilenkin系统Féjer平均的发散性问题，通过新的Vilenkin多项式构造，证明任意零测集上存在L_p函数使其Féjer平均发散，给出了显式构造的替代证明方法。

Ph.D. Thesis-Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series

In this PhD thesis we discuss, develop and apply this fascinating theory connected to modern harmonic analysis. In particular we make new estimations of Vilenkin-Fourier coefficients and prove some new results concerning boundedness of maximal operators of partial sums. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of the partial sums is valid and develop new methods to prove Hardy type inequalities for the partial sums with respect to the Vilenkin systems. We also do the similar investigation for the Fejér means. Furthermore, we…

On the convergence of partial sums with respect to Vilenkin system on the martingale Hardy spaces

In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for the modulus of continuity of $f\in H_{p}$ martingales, which provide convergence of subsequences of partial sums on the martingale Hardy spaces. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

On the Partial Sums and Marcinkiewicz and Fejér Means on the One- and Two-dimensional One-parameter Martingale Hardy Spaces

In this PhD thesis we are dealing with convergence and summability of partial sums, Fejér and Marcinkiewicz means with respect to one- and two-dimensional Walsh-Fourier series on the martingale Hardy spaces. This thesis is focus to achieve the following main results: To find estimation of convergence and divergence of the subsequences of partial sums of the one-dimensional Walsh-Fourier series on the martingale Hardy spaces $H_p(G)$, when $0<p\leq1$. To find necessary and sufficient conditions in terms of modulus of continuity of martingale Hardy spaces, for which subsequences of partial sums …

Bounded operators on Martingale Hardy spaces

The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are p…