Ushangi Goginava

Almost everywhere convergence of ( C , α)-means of cubical partial sums of d -dimensional Walsh-Fourier series

In the paper we prove that the maximal operator of the (C, α)-means of cubical partial sums of d- dimensional Walsh-Fourier series is of weak type (1,1). Moreover, the (C, α)-means σnαf of the function f ∈ L1 converge a.e. to f as n → ∞.

On the Uniform Convergence of Walsh-Fourier Series

We study the uniform convergence of Walsh-Fourier series of functions on the generalized Wiener class BV (p(n)↑∞)

Uniform convergence of Cesàro means of negative order of double Walsh--Fourier series

In this paper we prove that if f ∈ Cw([0, 1]2) and the function f is bounded partial p- variation for some p ∈ [1, + ∞) then the double Walsh-Fourier series of the function f is uniformly (C;-α,-β) summable (α + β 0) in the sense of Pringsheim. If α + β ≥ 1/p then there exists a continuous function f0 of bounded partial p-variation on [0, 1]2 such that the Cesaro (C;-α,-β) means σn,m-α,-β (f0, 0, 0) of the double Walsh-Fourier series of f0 diverge over cubes.

Strong Approximation by Marcinkiewicz Means of Two-dimensional Walsh-Fourier Series

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh-Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh-Fourier series of the continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

Maximal operator of the Fejér means of triangular partial sums oftwo-dimensional Walsh–Fourier series

Abstract. It is proved that the maximal operator of the triangular-Fejér-means of a two-dimensional Walsh–Fourier series is bounded from the dyadic Hardy space to for all and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular-Fejér-means of a function converge a.e. to . The maximal operator is bounded from the Hardy space to the space weak- and is not bounded from the Hardy space to the space .

Approximation properties of (C, α) means of double walsh-Fourier series

We study the rate of Lp approximation by Ces⦏ro means of the quadratic partial sums of double Walsh-Fourier series of functions from Lp.

Almost everywhere summability of multiple Walsh–Fourier series

Abstract We prove that for the N-dimensional Walsh–Fourier series the maximal operator of the Marcinkiewicz means is of weak type (1,1). Moreover, the Marcinkiewicz means σnf of the function f∈L1 converge a.e. to f as n→∞.

On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f ∈ Cϱ(⌊0, 1⌋N) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,−α) summable (α1 +...+ αN < 1, αi > 0, i = 1,...,N) in the sense of Pringsheim. If α1 +...+ αN = 1, αi > 0, i = 1,2,...,N, then there exists a continuous function f0 of bounded partial variation on [0, 1]N such that the Cesàro (C,−α) means σm−α(f0,Õ) of the N-dimensional Walsh-Fourier series of f0 diverge over cubes.

CONVERGENCE IN MEASURE OF LOGARITHMIC MEANS OF DOUBLE WALSH-FOURIER SERIES

Abstract The main aim of this paper is to prove that the logarithmic means of the double Walsh–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿log 𝐿(𝐼2), the set of functions for which quadratic logarithmic means of the double Walsh–Fourier series converge in measure is of first Baire category.

Convergence of double Fourier series and generalized Λ-variation

Abstract. The paper introduces a new concept of Λ-variation of bivariate functions and investigates its connection with the convergence of double Fourier series.