Mean Cesàro-type summability of Fourier-Neumann series
- Ciaurri, Ó. 1
- Stempak, K. 2
- Varona, J.L. 1
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1
Universidad de La Rioja
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2
Wrocław University of Technology
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ISSN: 0081-6906
Año de publicación: 2005
Volumen: 42
Número: 4
Páginas: 413-430
Tipo: Artículo
Otras publicaciones en: Studia Scientarum Mathematicarum Hungarica
Resumen
Let Jνv be the Bessel function of order νv. For α > -1, the functions x-α-1 Jα+2n+1(x), n = 0, 1, 2 ..., form an orthogonal system in L2(x2α+1 dx), but the span of such functions, is not dense in this space. For a function f, let Skαf denote the kth partial sum of the Fourier-Neumann series of f. In this paper we provide the minimal conditions on a real γ and 1 < p < ∞, for which the means Rnαf = λ oSoα + ⋯ +λ n Snαf,/λ o +⋯ +λn, λ k = 2(α + 2k + 2), are uniformly bounded in the spaces LP (x2(α+γ)+1 dx). Clearly, the convergence Rnαf → f holds only for functions from the closure of the linear span of the orthogonal system in these spaces. As a byproduct of the main result, we obtain a characterization of the closure of the span in terms of functions whose modified Hankel transforms of order α are supported on the interval [0,1]. © 2005 Akadémiai Kiadó, Budapest.