Strong convergence theorem for Walsh-Kaczmarz-Fejér means

Almost everywhere convergence of Fejér means of two-dimensional triangular Walsh-Fourier series

In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or $(C,1)$) means of the triangle two variable Walsh-Fourier series of $L^1$ functions. Namely, we prove the a.e. convergence $σ_n^{\bigtriangleup}f = \frac{1}{n}\sum_{k=0}^{n-1}S_{k, n-k}f\to f$ ($n\to\infty$) for each integrable two-variable function $f$.

Weighted maximal operators of Fejér means of Walsh-Fourier series in the martingale Hardy space $H_{1/2}$

In this paper we derive the restricted weighted maximal operator, defined by ${\sup }_{k\in \mathbb{N}}\left(\left\vert σ_{k}F\right\vert/A^2_k\right)$ of Fejér means of Walsh-Fourier series and prove that the it is bounded from the martingale Hardy space $H_{1/2}(G)$ to the Lebesgue space $L_{1/2}(G).$ The sharpness of this result is also proved. As a consequence we obtain some new and and well-know results.

Some new restricted maximal operators of Fejér means of Walsh-Fourier series in the space $H_{1/2}$

In this paper we derive the maximal subspace of natural numbers $\left\{n_{k}:k\geq 0\right\}$, such that the restricted maximal operator, defined by $\sup_{k\in \mathbb{N}}\left\vert σ_{n_{k}}F \right\vert$ on this subspace of Fejér means of Walsh-Fourier series is bounded from the martingale Hardy space $H_{1/2}$ to the Lebesgue space $L_{1/2}$. The sharpness of this result is also proved.

Restricted summability of the multi-dimensional Cesàro means of Walsh-Kaczmarz-Fourier series

The properties of the maximal operator of the $(C,α)$-means ($α=(α_1,\ldots,α_d)$) of the multi-dimensional Walsh-Kaczmarz-Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from $H_p^γ$ to $L_p$ for $p_0< p $ ($p_0=\max{1/(1+α_k): k=1,\ldots, d}$) and is of weak type $(1,1)$. As a corollary we get the theorem of Simon \cite{S1} on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. At the endpoint case $p=p_0$, we show that the maximal operator $σ^{κ,α,*}_L$ is not bounded fro…

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 …

Walsh-Kaczmarz-Fejér核的精确$L_1$-范数上界

The exact $L_1$-norm supremum of Walsh--Kaczmarz--Fejér kernels

本文研究了Walsh-Kaczmarz系统Fejér核的$L_1$范数上界，通过精确计算和分析得出其全局上界为71/50。