Note on Fourier series

L. S. Bosanquet; A. C. Offord

Displaying similar documents to “Note on Fourier series”

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.

Regularity of some nonlinear quantities on superharmonic functions in local Herz-type Hardy spaces.

Dashan Fan, Shanzhen Lu, Dachun Yang (1998)

Publicacions Matemàtiques

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In this paper, the authors introduce a kind of local Hardy spaces in R associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.

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...

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

Ferenc Weisz (1996)

Studia Mathematica

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The martingale Hardy space $H_{p} ({[0, 1)}^{2})$ and the classical Hardy space $H_{p} (^{2})$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_{p}$ to $L_{p}$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified...

The strong summability of Fourier series

G. Hardy, J. Littlewood (1935)

Fundamenta Mathematicae

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Continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator.

Zongguang Liu, Guozhen Lu, Shanzhen Lu (2003)

Publicacions Matemàtiques

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In this paper, we obtain some strong and weak type continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator on Hardy spaces, Hardy type spaces and weak Hardy type spaces.

On two extensions of the Hardy-Landau theorem

T. Varadarajan (1961)

Colloquium Mathematicae

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Dedekind and Hardy sums

R. Sitaramachandrarao (1987)

Acta Arithmetica

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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...

Two-parameter Hardy-Littlewood inequality and its variants

Chang-Pao Chen, Dah-Chin Luor (2000)

Studia Mathematica

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Let s* denote the maximal function associated with the rectangular partial sums $s_{m n} (x, y)$ of a given double function series with coefficients $c_{j k}$ . The following generalized Hardy-Littlewood inequality is investigated: ${| | s * | |}_{p, μ} \leq C_{p, α, β} {Σ_{j = 0}^{\infty} Σ_{k = 0}^{\infty} {(j ̅)}^{p - α - 2} {(k ̅)}^{p - β - 2} {| c_{j k} |}^{p}}^{1 / p}$ , where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{j k}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property...

$L^{p}$ -behaviour of the integral means of analytic functions

M. Mateljević, M. Pavlović (1984)

Studia Mathematica

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