-summability of Fourier series

Weisz, Ferenc

doi:10.1023/b:amhu.0000028241.87331.c5

Cited by 67 publications

(42 citation statements)

References 30 publications

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“…In the endpoint case p = 1/2 two positive results were showed. Weisz [22] proved that σ * is bounded from the Hardy space H 1/2 to the space weak-L 1/2 . In 2008 Goginava [8] proved that the maximal operatorσ * In 1948 Šneider [16] introduced the Walsh-Kaczmarz system and showed that the inequality lim sup n→∞ D κ n (x) log n C > 0 holds a.e.…”

Section: Introductionmentioning

confidence: 99%

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On the maximal operator of Walsh-Kaczmarz-Fejér means

Гогинава

Nagy

2011

Czech Math J

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Section: Introductionmentioning

confidence: 99%

“…Gát's result was generalized by Simon [14], who showed that the maximal operator σ κ, * is of type (H p , L p ) for p > 1/2. In the endpoint case p = 1/2 the first author [6] proved that maximal operator is not of type (H 1/2 , L 1/2 ) and Weisz [22] showed that the maximal operator is of weak type (H 1/2 , L 1/2 ).…”

Section: Introductionmentioning

confidence: 99%

On the maximal operator of Walsh-Kaczmarz-Fejér means

Гогинава

Nagy

2011

Czech Math J

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“…The next sharper result, which is well known for the trigonometric, Walsh-and Vilenkin systems, is due to the author [66].…”

Section: Convergence Of Ciesielski-fourier Seriesmentioning

confidence: 85%

“…For the multi-dimensional versions of the theorems presented in this paper see [58,63,68,60,66,65,70,61]. To shorten the paper we do not give these extensions here.…”

Section: 4mentioning

confidence: 99%

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Results on spline-Fourier and Ciesielski-Fourier series

Weisz

2006

Approximation and Probability

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Abstract. Some recent results on spline-Fourier and Ciesielski-Fourier series are summarized. The convergence of spline Fourier series and the basis properties of the spline systems are considered. Some classical topics, that are well known for trigonometric and Walsh-Fourier series, are investigated for Ciesielski-Fourier series, such as inequalities for the Fourier coefficients, convergence a.e. and in norm, Fejér and θ-summability, strong summability and multipliers. The connection between Fourier series and Hardy spaces is studied.1. Introduction. It is known that the spline Fourier series of f ∈ L p converges a.e. and in L p norm to f , whenever 1 ≤ p < ∞. The maximal operator of the partial sums with respect to the spline (or unbounded Ciesielski) systems of order (m, k) is bounded from the Hardy space H p to L p (1/(m − k + 2) < p < ∞) and is of weak type (1, 1). If 1 < p < ∞ then H p is equivalent to L p . Moreover, the spline systems are unconditional and equivalent bases to the Haar system in H p (1/(m − k + 2) < p < ∞).We investigate also the bounded Ciesielski systems, which can be obtained from the spline systems in the same way as the Walsh system from the Haar. Some results, that are well known for trigonometric and Walsh-Fourier series, are extended to CiesielskiFourier series. Paley and Hardy-Littlewood type inequalities are shown for CiesielskiFourier coefficients. If f ∈ L p (1 < p < ∞) then the Ciesielski-Fourier series of f converges

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Herz spaces and summability of Fourier transforms

Feichtinger

Weisz

2008

Mathematische Nachrichten

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A general summability method is considered for functions from Herz spaces Kαp,r (ℝd ). The boundedness of the Hardy–Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ ‐means σθ T f is also bounded on the corresponding Herz spaces and σθ T f → f a.e. for all f ∈ K–d /p p,∞ (ℝd ). Moreover, σθ T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K–d /p p,∞ (ℝd ) if and only if the Fourier transform of θ is in the Herz space Kd /p p ′,1 (ℝd ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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-summability of Fourier series

Cited by 67 publications

References 30 publications

On the maximal operator of Walsh-Kaczmarz-Fejér means

On the maximal operator of Walsh-Kaczmarz-Fejér means

Results on spline-Fourier and Ciesielski-Fourier series

Herz spaces and summability of Fourier transforms

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