I am a mathematician interested in Fourier analysis of one and multivariable functions mainly with respect to the trigonometric, Walsh and Walsh-like systems.
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 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.
In this paper, we give a description of points at which the strong means of VilenkinFourier serie... more In this paper, we give a description of points at which the strong means of VilenkinFourier series converge.
For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): ... more For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \sum\limits_{j = 0}^\infty {\left| {{n_j} - {n_{j + 1}}} \right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) < ∞, the subsequence of quadratic partial sums $$S{_n^{\square}}_A\left( f \right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) < ∞ and a function f ∈ φ(L)(I2) for which $$\text{sup} _A|S{_n^{\square}}_A\left( {{x^1},{x^2};f} \right)| = \infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.
Izvestiâ Akademii nauk Armânskoj SSR. Matematika, Jul 18, 2022
The element of the Walsh system, that is the Walsh functions map from the unit interval to the se... more The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.
Journal of Contemporary Mathematical Analysis, Jul 1, 2019
In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional func... more In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means $$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$$ holds, where a(n) is a lacunary sequence of positive integers.
In this paper, we give a description of points at which the strong means of VilenkinFourier serie... more In this paper, we give a description of points at which the strong means of VilenkinFourier series converge.
For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): ... more For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \sum\limits_{j = 0}^\infty {\left| {{n_j} - {n_{j + 1}}} \right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) < ∞, the subsequence of quadratic partial sums $$S{_n^{\square}}_A\left( f \right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) < ∞ and a function f ∈ φ(L)(I2) for which $$\text{sup} _A|S{_n^{\square}}_A\left( {{x^1},{x^2};f} \right)| = \infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.
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