Several Dimensional θ-Summability and Hardy Spaces

Journal of Mathematical Analysis and Applications

2008

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A general summability method, the so-called θ -summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1, 1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σ θ T f → f over a cone-like set a.e. for all f ∈ L 1 (R d ). Moreover, σ θ T f (x) converges to f (x) over a cone-like set at each Lebesgue point of f ∈ L 1 (R d ) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.

Section: Introductionmentioning

confidence: 90%

“…The so-called θ -summation is a general method of summation generated by one single function θ and it is intensively studied in the literature (see e.g. Butzer and Nessel [3], Trigub and Belinsky [19], Bokor, Schipp, Szili and Vértesi [2,13,16,17] and Weisz [5,6,[22][23][24]). …”

Section: Introductionmentioning

confidence: 99%

Herz spaces and restricted summability of Fourier transforms and Fourier series

Journal of Mathematical Analysis and Applications

2008

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“…This convergence has been investigated in a great number of papers (e.g. in Marcinkiewicz and Zygmund [18], Zygmund [32], Weisz [27,29,30] …”

Section: ≤1mentioning

confidence: 99%

“…The θ-summation was considered in a great number of papers and books, such as Butzer and Nessel [3], Bokor, Schipp, Szili, and Vértesi [19,2,20,24,23], Weisz [28,29,31,30] and Feichtinger and Weisz [5,6].…”

mentioning

confidence: 99%

Herz spaces and summability of Fourier transforms

Feichtinger

Mathematische Nachrichten

2008

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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)

“…A general method of summation, the so called θ -summation method, which is generated by a single function θ is studied intensively in the literature (see, e.g., Butzer and Nessel [4], Trigub and Belinsky [29] and Weisz [30,31] and the references therein). If the Fourier transform of θ is integrable then the norm convergence results just mentioned hold for the θ -summation method, too (see Butzer and Nessel [4], Trigub and Belinsky [29] or Feichtinger and Weisz [10]).…”

mentioning

confidence: 99%

Pointwise Summability of Gabor Expansions

2008

J Fourier Anal Appl

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A general summability method, the so-called θ -summability method is considered for Gabor series. It is proved that if the Fourier transform of θ is in a Herz space then this summation method for the Gabor expansion of f converges to f almost everywhere when f ∈ L 1 or, more generally, when f ∈ W (L 1 , ∞ ) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the θ -means of the Gabor expansion are obtained. Hardy-Littlewood type maximal functions are introduced and some inequalities are proved for these.