Gyorgy Gat
University of Debrecen, Institute of Mathematics, Faculty Member
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I am a mathematician interested in Fourier analysis of one and multivariable functions mainly with respect to the trigonometric, Walsh and Walsh-like systems.edit
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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... 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.
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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... 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.
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In this paper, we give a description of points at which the strong means of VilenkinFourier series converge.
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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... 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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We prove that the maximal operator of the (C,?n)-means of the one dimensional Vilenkin-Fourier series is of weak type(L1, L1). Moreover, we prove the almost everywhere convergence of the (C,?n) means of integrable functions (i.e. ? ?n n f... more
We prove that the maximal operator of the (C,?n)-means of the one dimensional Vilenkin-Fourier series is of weak type(L1, L1). Moreover, we prove the almost everywhere convergence of the (C,?n) means of integrable functions (i.e. ? ?n n f ? f ), where n ? N?,q and n ? 1 for f ? L1(Gm), Gm is a bounded Vilenkin group, for every sequence ? = (?n), 0 < ?n < 1.
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The main aim of this article is to demonstrate the difference of the trigonometric and the Walsh system with respect to the behaviour of the maximal function of the Fejér kernels. Moreover, properties (positivity among others) of the... more
The main aim of this article is to demonstrate the difference of the trigonometric and the Walsh system with respect to the behaviour of the maximal function of the Fejér kernels. Moreover, properties (positivity among others) of the Walsh logarithmic kernels are also investigated.
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The main aim of this paper is to prove that the (𝐶, α)-means of quadratic partial sums of double Walsh–Kaczmarz–Fourier series are of weak type (1, 1) and of type (𝑝, 𝑝) for all 1 < 𝑝 ≤ ∞ (0 < α < 1). Moreover, these (𝐶, α)-means... more
The main aim of this paper is to prove that the (𝐶, α)-means of quadratic partial sums of double Walsh–Kaczmarz–Fourier series are of weak type (1, 1) and of type (𝑝, 𝑝) for all 1 < 𝑝 ≤ ∞ (0 < α < 1). Moreover, these (𝐶, α)-means converge to 𝑓 almost everywhere for any integrable function 𝑓.
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ABSTRACT In 1992, Móricz, Schipp and Wade [6] proved the a.e. convergence of the double (Italic&gt;C,1) means of the Walsh--Fourier series &amp;sgr;Italic&gt;n f → Italic&gt;f (min (Italic&gt;n1, Italic&gt;n2)→∞,... more
ABSTRACT In 1992, Móricz, Schipp and Wade [6] proved the a.e. convergence of the double (Italic&gt;C,1) means of the Walsh--Fourier series &amp;sgr;Italic&gt;n f → Italic&gt;f (min (Italic&gt;n1, Italic&gt;n2)→∞, Italic&gt;n=(Italic&gt;n1,Italic&gt;n2) ∈ Bold&gt;N2) for functions in Italic&gt;L log+ Italic&gt;L ([0,1)2). This result for bounded Vilenkin groups is generalized by Weisz [10]. We show that these results can not be improved with respect to two-dimensional bounded Vilenkin groups (not only the two-dimensional Walsh group). We prove that for all measurable functions &amp;dgr; : [0,+∞) → [0,+∞), lim t → ∞ &amp;dgr;(Italic&gt;t) = 0, Italic&gt;GmxItalic&gt;Gm&amp;apos; two-dimensional bounded Vilenkin group we have an f ∈ Italic&gt;L log+ Italic&gt;L&amp;dgr;(Italic&gt;L) such that &amp;sgr;Italic&gt;nf does not converge to Italic&gt;f a.e. (in the Pringsheim sense).
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... n-1 mn ~1 Ia - j=mn -qn =: ia(t)r - t)(n + E)dt. h It is easy to get (6) ,.~.xt) }dt = B I, Define Ak (ke J := {a,a + 1,... ,ft}) in the fonowing way: := (fl - a) / a(z)~bq,,+,M,,+l+...+q,M,(z)dz (k EJ). Ak I~(y,k) Thus IAkl =< 1... more
... n-1 mn ~1 Ia - j=mn -qn =: ia(t)r - t)(n + E)dt. h It is easy to get (6) ,.~.xt) }dt = B I, Define Ak (ke J := {a,a + 1,... ,ft}) in the fonowing way: := (fl - a) / a(z)~bq,,+,M,,+l+...+q,M,(z)dz (k EJ). Ak I~(y,k) Thus IAkl =< 1 (ke J) and in the case of q < Mn+l, ~ At, = 0 holds. k6.J ...
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The (Nörlund) logarithmic means of the Fourier series of the integrable function f is: 1 ln n−1∑ k=1 Sk(f) n − k, where ln:= n−1∑ k=1
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In this paper we give a common generalization of the Walsh, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions of continuous irreducible unitary... more
In this paper we give a common generalization of the Walsh, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions of continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We introduce the notion of the modulus of continuity on Vilenkin spaces, the concept of the best approximation by Vilenkin-like polynomials. We prove a Jackson type theorem. Denote by N the set of natural numbers, P the set of positive integers, respectively. Denote m := (mk : k ∈ N) a sequence of positive integers such that mk ≥ 2, k ∈ N and Gmk a set of cardinality mk. Suppose that each (coordinate) set has the discrete topology and measure μk which maps every singleton of Gmk to 1 mk (μk(Gmk) = 1), k ∈ N. Let Gm be the compact set formed by the complete direct product of Gmk with the product of the topologies a...
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We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛𝑓... more
We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛𝑓 converges to 𝑓 almost everywhere.
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We prove that the maximal operator of the (𝐶, α)-means of quadratical partial sums of double Vilenkin–Fourier series is of weak type (1,1). Moreover, the (𝐶, α)-means of a function 𝑓 ∈ 𝐿1 converge a.e. to 𝑓 as 𝑛 → ∞.