Cesàro summability of one- and two-dimensional trigonometric-Fourier series

Ferenc Weisz

Displaying similar documents to “Cesàro summability of one- and two-dimensional trigonometric-Fourier series”

Fejér means of two-dimensional Fourier transforms on $H_{p} (ℝ \times ℝ)$

Ferenc Weisz (1999)

Colloquium Mathematicae

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The two-dimensional classical Hardy spaces $H_{p} (ℝ \times ℝ)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_{p} (ℝ \times ℝ)$ to $L_{p} (ℝ^{2})$ (1/2 < p ≤ ∞) and is of weak type $(H_{1}^{} (ℝ \times ℝ), L_{1} (ℝ^{2}))$ where the Hardy space $H_{1} (ℝ \times ℝ)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_{1}^{(} ℝ \times ℝ)$ ⊃ $L l o g L (ℝ^{2})$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_{p} (ℝ \times ℝ)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_{p} (ℝ \times ℝ)$ , the...

Necessary conditions for the $L^{p}$ -convergence $(0 < p < 1)$ of single and double trigonometric series

Xhevat Z. Krasniqi, Péter Kórus, Ferenc Móricz (2014)

Mathematica Bohemica

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We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the $L^{p}$ -metric, where $0 < p < 1$ . The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in $H^{p}$ and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the $L^{p}$ -metric, where $0 < p < 1$ .

On $[F, d_{n}]$ summability of Fourier series

P. L. Sharma, V. Prakash, S. K. Varma (1985)

Matematički Vesnik

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On the maximal Fejér operator for double Fourier series of functions in Hardy spaces

Ferenc Móricz (1995)

Studia Mathematica

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We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1, 0)} (^{2})$ , $H^{(0, 1)} (^{2})$ , or $H^{(1, 1)} (^{2})$ . We prove that the maximal Fejér operator is bounded from $H^{(1, 0)} (^{2})$ or $H^{(0, 1)} (^{2})$ into weak- $L^{1} (^{2})$ , and also bounded from $H^{(1, 1)} (^{2})$ into $L^{1} (^{2})$ . These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1} l o g^{+} L (^{2})$ , $L^{1} {(l o g^{+} L)}^{2} (^{2})$ , and $L^{μ} (^{2})$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate...

On the maximal operator of Walsh-Kaczmarz-Fejér means

Ushangi Goginava, Károly Nagy (2011)

Czechoslovak Mathematical Journal

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In this paper we prove that the maximal operator ${\tilde{σ}}^{κ, *} f : = sup_{n \in ℙ} \frac{| σ_{n}^{κ} f |}{{log}^{2} (n + 1)},$ where $σ_{n}^{κ} f$ is the $n$ -th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_{1 / 2} (G)$ to the space $L_{1 / 2} (G) .$

On the Fejér means of bounded Ciesielski systems

Ferenc Weisz (2001)

Studia Mathematica

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We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space $H_{p}$ to $L_{p}$ if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1,1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if...

Multi-dimensional Fejér summability and local Hardy spaces

Ferenc Weisz (2009)

Studia Mathematica

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It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space $W (h_{p}, ℓ_{\infty})$ to $W (L_{p}, ℓ_{\infty})$ . This implies the almost everywhere convergence of the Fejér means in a cone for all $f \in W (L ₁, ℓ_{\infty})$ , which is larger than $L ₁ (ℝ^{d})$ .

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

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Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of $L_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

On the Nörlund means of Vilenkin-Fourier series

István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)

Czechoslovak Mathematical Journal

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We prove and discuss some new $(H_{p}, L_{p})$ -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients ${q_{k} : k \geq 0}$ . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of...

On $| V, λ |$ summability of a factored Fourier series

P. L. Sharma, R. K. Jain (1970)

Matematički Vesnik

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Norm convergence of Fejér means of two-dimensional Walsh-Fourier series

Ushangi Goginava (2011)

Banach Center Publications

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The main aim of this paper is to prove that there exists a martingale $f \in H_{1 / 2}$ such that the Fejér means of the two-dimensional Walsh-Fourier series of f is not uniformly bounded in the space weak- $L_{1 / 2}$ .

Walsh-Marcinkiewicz means and Hardy spaces

Károly Nagy, George Tephnadze (2014)

Open Mathematics

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The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H p, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H p, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.