Mean Cesàro-type summability of Fourier-Neumann series

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.