I am a mathematician interested in Fourier analysis of one and multivariable functions mainly with respect to the trigonometric, Walsh and Walsh-like systems.
The boundedness of maximal operators of subsequences of (C, αn)-means of partial sums of Walsh-Fo... more The boundedness of maximal operators of subsequences of (C, αn)-means of partial sums of Walsh-Fourier series from the Hardy space Hp into the space Lp is studied.
The main aim of this paper is to prove that the logarithmic means of the double Walsh-Fourier ser... more 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 L log L(I 2 ), the set of functions for which quadratic logarithmic means of the double Walsh-Fourier series converge in measure is of first Baire category.
The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Vil... more The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Vilenkin-Fourier series of two variable integrable functions on two dimensional bounded Vilenkin groups. That is, let a = (a1, a2) : N → N 2 such that aj(n + 1) ≥ δ sup k≤n aj(n) (j = 1, 2, n ∈ N) for some δ > 0 and a1(+∞) = a2(+∞) = +∞. Then for each integrable function f ∈ L 1 (G 2 m ) we have the a.e. relation limn→∞ σ a 1 (n),a 2 (n) f = f . It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the twodimensional Vilenkin-Fejér means of integrable functions which was proved earlier by Weisz [13] and Blahota and the author [2] independently. The trigonometric and Walsh's analogue of the main result see Gát [6],
The aim of this work is to generalize the more than 60 year old celebrated result of Marcinkiewic... more The aim of this work is to generalize the more than 60 year old celebrated result of Marcinkiewicz and Zygmund on the convergence of the two-dimensional restricted (C, 1) means of trigonometric Fourier series. They proved for any integrable function f ∈ L 1 (T 2 ) the a.e. convergence where > 1 is fixed constant. That is, the set of indices (n 1 , n 2 ) remains in some positive cone around the identical function. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets (of the set of indices) in order to preserve this convergence property.
It is proved that the operators σ △ n of the triangular-Fejérmeans of a two-dimensional Walsh-Fou... more It is proved that the operators σ △ n of the triangular-Fejérmeans of a two-dimensional Walsh-Fourier series are uniformly bounded from the dyadic Hardy space Hp to Lp for all 4/5 < p ≤ ∞.
Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres
Ukraïnsʹkij matematičnij žurnal, Mar 11, 2021
UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variabl... more UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type . Moreover, the -means of the function converge a.e. to for , where is the Walsh group for some sequences .
In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-di... more In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation ( From the theorem of Getsadze [6] it follows that the space L log L can not be enlarged with preserving this strong summability property.
In this paper we investigate the almost everywhere convergence of two dimensional Marcinkiewicz-l... more In this paper we investigate the almost everywhere convergence of two dimensional Marcinkiewicz-like means of two variable integrable functions which is given by t α n f = 1 n n−1 ∑ k=0 S α 1 (|n|, k),α 2 (|n|, k) f (M |n| ≤ n < M |n|+1) and give a sufficient condition for functions α : N 2 → N 2 in order to have the a.e. relation t α n f → f for all f ∈ L 1 (G 2 m) with respect to two dimensional bounded Vilenkin-like systems. Finally, we give an application of the main result with respect to triangular summability of Vilenkin-like-Fourier series.
Facta universitatis. Series electronics and energetics, 2008
The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dim... more The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Vilenkin-Fourier series is of weak type (1, 1). Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Vilenkin-Fourier series is of weak type (1, 1), provided that the supremum in the maximal operator is taken over special indices. The set of Vilenkin polynomials is dense in L 1 , so by the well-known density argument the logarithmic means t 2 n (f) converge a.e. to f for all integrable function f .
In the present paper, we prove the almost everywhere convergence and divergence of subsequences o... more In the present paper, we prove the almost everywhere convergence and divergence of subsequences of Cesàro means with zero tending parameters of Walsh-Fourier series.
Izvestiâ Akademii nauk Armânskoj SSR. Matematika, Jul 18, 2022
The element of the Walsh system, that is the Walsh functions map from the unit interval to the se... more The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.
Journal of Fourier Analysis and Applications, Oct 16, 2017
In 1987 Harris proved [11]-among others-that for each 1 ≤ p < 2 there exists a two-dimensional fu... more In 1987 Harris proved [11]-among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ 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 f = 1 n n−1 k=0 S k,n−k f → f (n → ∞) for each integrable two-variable function f. 2010 Mathematics Subject Classification. 42C10.
In 1987 Harris proved-among others-that for each 1 ≤ p < 2 there exists a two-dimensional functio... more In 1987 Harris proved-among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ L p such that its triangular partial sums S 2 A f of Walsh-Fourier series does not converge almost everywhere. In this paper we prove that subsequences of triangular partial sums S n A M A f, n A ∈ {1, 2, ..., m A − 1} on unbounded Vilenkin groups converge almost everywhere to f for each function f ∈ L 2 .
We consider sequences of compact bounded linear operators U n : L p (0, 1) → L p (0, 1) with cert... more We consider sequences of compact bounded linear operators U n : L p (0, 1) → L p (0, 1) with certain convergence properties. Several divergence theorems for the multiple sequences of tensor products of these operators are proved. These theorems in particular imply that L log d−1 L is the optimal Orlicz space guaranteeing almost everywhere summability of rectangular partial sums of multiple Fourier series in general orthogonal systems.
Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series
Journal of Contemporary Mathematical Analysis, Jul 1, 2019
In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional func... more In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means $$S_{2^A}^\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence $$S_{a(n)}^\Delta (f) \rightarrow f$$ holds, where a(n) is a lacunary sequence of positive integers.
We study approximation by rectangular partial sums of double Fourier series on unbounded Vilenkin... more We study approximation by rectangular partial sums of double Fourier series on unbounded Vilenkin groups in the spaces C and L1. From these results we obtain criterions of the uniform convergence and L-convergence of double Vilenkin-Fourier series. We also prove that these results are sharp.
Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric sys... more Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems established by Rodin in 1990. We prove, that if the growth of a function Φ(t) : [0, ∞) → [0, ∞) is bigger than the exponent, then the strong Φ-summability of a Walsh-Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the author of this paper.
The boundedness of maximal operators of subsequences of (C, αn)-means of partial sums of Walsh-Fo... more The boundedness of maximal operators of subsequences of (C, αn)-means of partial sums of Walsh-Fourier series from the Hardy space Hp into the space Lp is studied.
The main aim of this paper is to prove that the logarithmic means of the double Walsh-Fourier ser... more 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 L log L(I 2 ), the set of functions for which quadratic logarithmic means of the double Walsh-Fourier series converge in measure is of first Baire category.
The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Vil... more The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Vilenkin-Fourier series of two variable integrable functions on two dimensional bounded Vilenkin groups. That is, let a = (a1, a2) : N → N 2 such that aj(n + 1) ≥ δ sup k≤n aj(n) (j = 1, 2, n ∈ N) for some δ > 0 and a1(+∞) = a2(+∞) = +∞. Then for each integrable function f ∈ L 1 (G 2 m ) we have the a.e. relation limn→∞ σ a 1 (n),a 2 (n) f = f . It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the twodimensional Vilenkin-Fejér means of integrable functions which was proved earlier by Weisz [13] and Blahota and the author [2] independently. The trigonometric and Walsh's analogue of the main result see Gát [6],
The aim of this work is to generalize the more than 60 year old celebrated result of Marcinkiewic... more The aim of this work is to generalize the more than 60 year old celebrated result of Marcinkiewicz and Zygmund on the convergence of the two-dimensional restricted (C, 1) means of trigonometric Fourier series. They proved for any integrable function f ∈ L 1 (T 2 ) the a.e. convergence where > 1 is fixed constant. That is, the set of indices (n 1 , n 2 ) remains in some positive cone around the identical function. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets (of the set of indices) in order to preserve this convergence property.
It is proved that the operators σ △ n of the triangular-Fejérmeans of a two-dimensional Walsh-Fou... more It is proved that the operators σ △ n of the triangular-Fejérmeans of a two-dimensional Walsh-Fourier series are uniformly bounded from the dyadic Hardy space Hp to Lp for all 4/5 < p ≤ ∞.
Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres
Ukraïnsʹkij matematičnij žurnal, Mar 11, 2021
UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variabl... more UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type . Moreover, the -means of the function converge a.e. to for , where is the Walsh group for some sequences .
In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-di... more In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation ( From the theorem of Getsadze [6] it follows that the space L log L can not be enlarged with preserving this strong summability property.
In this paper we investigate the almost everywhere convergence of two dimensional Marcinkiewicz-l... more In this paper we investigate the almost everywhere convergence of two dimensional Marcinkiewicz-like means of two variable integrable functions which is given by t α n f = 1 n n−1 ∑ k=0 S α 1 (|n|, k),α 2 (|n|, k) f (M |n| ≤ n < M |n|+1) and give a sufficient condition for functions α : N 2 → N 2 in order to have the a.e. relation t α n f → f for all f ∈ L 1 (G 2 m) with respect to two dimensional bounded Vilenkin-like systems. Finally, we give an application of the main result with respect to triangular summability of Vilenkin-like-Fourier series.
Facta universitatis. Series electronics and energetics, 2008
The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dim... more The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Vilenkin-Fourier series is of weak type (1, 1). Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Vilenkin-Fourier series is of weak type (1, 1), provided that the supremum in the maximal operator is taken over special indices. The set of Vilenkin polynomials is dense in L 1 , so by the well-known density argument the logarithmic means t 2 n (f) converge a.e. to f for all integrable function f .
In the present paper, we prove the almost everywhere convergence and divergence of subsequences o... more In the present paper, we prove the almost everywhere convergence and divergence of subsequences of Cesàro means with zero tending parameters of Walsh-Fourier series.
Izvestiâ Akademii nauk Armânskoj SSR. Matematika, Jul 18, 2022
The element of the Walsh system, that is the Walsh functions map from the unit interval to the se... more The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.
Journal of Fourier Analysis and Applications, Oct 16, 2017
In 1987 Harris proved [11]-among others-that for each 1 ≤ p < 2 there exists a two-dimensional fu... more In 1987 Harris proved [11]-among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ 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 f = 1 n n−1 k=0 S k,n−k f → f (n → ∞) for each integrable two-variable function f. 2010 Mathematics Subject Classification. 42C10.
In 1987 Harris proved-among others-that for each 1 ≤ p < 2 there exists a two-dimensional functio... more In 1987 Harris proved-among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ L p such that its triangular partial sums S 2 A f of Walsh-Fourier series does not converge almost everywhere. In this paper we prove that subsequences of triangular partial sums S n A M A f, n A ∈ {1, 2, ..., m A − 1} on unbounded Vilenkin groups converge almost everywhere to f for each function f ∈ L 2 .
We consider sequences of compact bounded linear operators U n : L p (0, 1) → L p (0, 1) with cert... more We consider sequences of compact bounded linear operators U n : L p (0, 1) → L p (0, 1) with certain convergence properties. Several divergence theorems for the multiple sequences of tensor products of these operators are proved. These theorems in particular imply that L log d−1 L is the optimal Orlicz space guaranteeing almost everywhere summability of rectangular partial sums of multiple Fourier series in general orthogonal systems.
Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series
Journal of Contemporary Mathematical Analysis, Jul 1, 2019
In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional func... more In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means $$S_{2^A}^\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence $$S_{a(n)}^\Delta (f) \rightarrow f$$ holds, where a(n) is a lacunary sequence of positive integers.
We study approximation by rectangular partial sums of double Fourier series on unbounded Vilenkin... more We study approximation by rectangular partial sums of double Fourier series on unbounded Vilenkin groups in the spaces C and L1. From these results we obtain criterions of the uniform convergence and L-convergence of double Vilenkin-Fourier series. We also prove that these results are sharp.
Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric sys... more Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems established by Rodin in 1990. We prove, that if the growth of a function Φ(t) : [0, ∞) → [0, ∞) is bigger than the exponent, then the strong Φ-summability of a Walsh-Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the author of this paper.
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Papers by Gyorgy Gat