Ferenc Móricz

Almost convergence of double sequences and strong regularity of summability matrices

A double sequence x = { x jk : j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if that is, the average value of { x jk } taken over any rectangle {( j, k ): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n . The notion of almost convergence for single sequences was introduced by Lorentz [ 1 ].

Cesàro summability of double Walsh-Fourier series

We introduce quasi-local operators (these include operators of Calder6n-Zygmund type), a hybrid Hardy space HO of functions of two variables, and we obtain sufficient conditions for a quasi-local maximal operator to be of weak type ( , 1) . As an application, we show that Cesatro means of the double Walsh-Fourier series of a function f converge a.e. when f belongs to HO. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all f E L1 provided summability takes place in some positive cone.

Tauberian conditions, under which statistical convergence follows from statistical summability (C,1)☆

Abstract J.A. Fridly and M.K. Khan have recently extended Hardys and Landaus Tauberian theorems to the case of statistical convergence, which was introduced by H. Fast in 1951. Let (x k : k=0,1,2,…) be a sequence of real or complex numbers and set σn:=(n+1)−1∑nk=0xk for n=0,1,2,… . We present necessary and sufficient conditions, under which st-limxk=L follows from st-limσn=L, where L is a finite number. If (xk) is a sequence of real numbers, then these are one-sided Tauberian conditions. If (xk) is a sequence of complex numbers, then these are two-sided Tauberian conditions. In particular, our conditions are satisfied if (xk) is statistically slowly decreasing (or increasing) in the case of real sequences; or if (xk) is statistically slowly oscillating in the case of complex sequences. Even these special sufficient conditions imply those given by Fridy and Khan.

On the convergence in a restricted sense of multiple series

AbstractПустьZd — множество на боров k=(k1, ...,kd) неотрицательных цел ых чиселkj,d≧1.d-кратный числовой ря д (8), т. е. ряд

Approximation by No¨rlund means of Walsh-Fourier series

\mathop \sum \limits_{k \geqq 1}

Ferenc Lukács type theorems in terms of the Abel-Poisson mean of conjugate series

ak прямоуголь ными частными суммамиsm=

Tauberian theorems for double sequences that are statistically summable (C,1,1)☆

\mathop \sum \limits_{1 \leqq k \leqq m}