2019
Том 71
№ 11

# Goginava U.

Articles: 5
Article (English)

### Strong summability of two-dimensional Vilenkin – Fourier series

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 340-352

We study the exponential uniform strong summability of two-dimensional Vilenkin – Fourier series. In particular, it is proved that the two-dimensional Vilenkin – Fourier series of a continuous function $f$ is uniformly strongly summable to a function $f$ exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

Article (English)

### Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 163-173

The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series.

Article (English)

### On the summability of double Walsh - Fourier series of functions of bounded generalized variation

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 490-507

The convergence of Cesaro means of negative order of double Walsh-Fourier series of functions of bounded generalized variation is investigated.

Article (English)

### On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 158–166

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator $σ^{α,κ,*}$ of the $(C, α)$-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space $H_p$ to the space $L_p$ for $p > 1 / (1 + α), \;0 < α ≤ 1$. Recently, Gát and Goginava have proved that this boundedness result does not hold if $p ≤ 1 / (1 + α)$. However, in the endpoint case $p = 1 / (1 + α )$, the maximal operator $σ^{α,κ,*}$ is bounded from the martingale Hardy space $H_{1/(1+α)}$ to the space weak- $L_{1/(1+α)}$. The main aim of this paper is to prove a stronger result, namely, that, for any $0 < p ≤ 1 / (1 + α)$, there exists a martingale $f ∈ H_p$ such that the maximal operator $σ^{α,κ,*} f$ does not belong to the space $L_p$.

Brief Communications (English)

### On the embedding of Waterman class in the class Hpω

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1557–1562

In this paper the necessary and sufficient condition for the inclusion of class ЛBV in the class Hpω is found.