Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series

Ferenc Weisz

Displaying similar documents to “Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series”

On the $L_{1}$ -convergence of Fourier series

S. Fridli (1997)

Studia Mathematica

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Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier...

Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Péter Simon, Ferenc Weisz (1997)

Studia Mathematica

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Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $(\sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} {| f ̂ (k, j) |}^{p} {(k j)}^{p - 2})^{1 / p} \leq C_{p} ∥ f ∥_{H_{* *}^{p}}$ (1/2 < p≤2) where f belongs to the Hardy space $H_{* *}^{p} (G_{m} \times G_{s})$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.

The strong summability of Fourier series

G. Hardy, J. Littlewood (1935)

Fundamenta Mathematicae

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Note on Fourier series

L. S. Bosanquet, A. C. Offord (1935)

Compositio Mathematica

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Two-parameter Hardy-Littlewood inequalities

Ferenc Weisz (1996)

Studia Mathematica

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The inequality (*) $(\sum_{| n | = 1}^{\infty} \sum_{| m | = 1}^{\infty} {| n m |}^{p - 2} {| f ̂ (n, m) |}^{p})^{1 / p} \leq C_{p} ∥ ƒ ∥_{H_{p}}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_{p}$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_{p}$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier...

Almost everywhere convergence of subsequence of logarithmic means of Walsh-Fourier series.

Goginava, Ushangi (2005)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

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Inequalities relative to two-parameter Vilenkin-Fourier coefficients

Ferenc Weisz (1991)

Studia Mathematica

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On the new convergence criteria for Fourier series of Hardy and Littlewood

Gilbert Walter Morgan (1935)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Almost everywhere convergence of Walsh Fourier series of $H^{1}$ -functions

N. Ladhawala, D. Pankratz (1976)

Studia Mathematica

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Weighted integrability and L¹-convergence of multiple trigonometric series

Chang-Pao Chen (1994)

Studia Mathematica

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We prove that if $c_{j k} \to 0$ as max(|j|,|k|) → ∞, and $\sum_{| j | = 0 \pm}^{\infty} \sum_{| k | = 0 \pm}^{\infty} {θ (| j |}^{⊤} {) ϑ (| k |}^{⊤}) | Δ_{12} c_{j k} | < \infty$ , then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and $\iint_{T ²} | s_{m n} (x, y) - f (x, y) | \cdot | ϕ (x) ψ (y) | d x d y \to 0$ as min(m,n) → ∞, where f(x,y) is the limiting function of the rectangular partial sums $s_{m n} (x, y)$ , (ϕ,θ) and (ψ,ϑ) are pairs of type I. A generalization of this result concerning L¹-convergence is also established. Extensions of these results to double series of orthogonal functions are also considered. These results can be extended to n-dimensional case. The aforementioned results generalize work of Balashov [1],...