Mean Cesàro-type summability of Fourier-Neumann series
Authors: Ciaurri Óscar , Stempak Krzysztof , Varona Juan L. 
Published: November 2005

Let \documentclass{aastex}
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\begin{document}
$J_\nu$
\end{document} be the Bessel function of order \documentclass{aastex}
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\begin{document}
$\nu$
\end{document}. For \documentclass{aastex}
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\begin{document}
$\alpha?-1$
\end{document}, the functions \documentclass{aastex}
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\begin{document}
$x^{-\alpha-1}J_{\alpha+2n+1}(x)$
\end{document}, \documentclass{aastex}
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\begin{document}
$n=0,1,2\dots$
\end{document}, form an orthogonal system in \documentclass{aastex}
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\begin{document}
$L^2(x^{2\alpha+1}\,dx)$
\end{document}, but the span of such functions is not dense in this space. For a function \documentclass{aastex}
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\begin{document}
$f$
\end{document}, let \documentclass{aastex}
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\begin{document}
$S_k^{\alpha}f$
\end{document} denote the $kth partial sum of the Fourier--Neumann series of \documentclass{aastex}
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\begin{document}
$f$
\end{document}. In this paper we provide the minimal conditions on a real \documentclass{aastex}
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\begin{document}
$\gamma$
\end{document} and \documentclass{aastex}
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\begin{document}
$1<p<\infty$
\end{document}, for which the means \documentclass{aastex}
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\begin{document}
$R_n^{\alpha}f = \frac{\lambda_0 S_0^{\alpha}f + \cdots + \lambda_n S_n^{\alpha}f}{\lambda_0 + \cdots + \lambda_n}$
\end{document}, \documentclass{aastex}
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\begin{document}
$\lambda_k = 2(\alpha+2k+2$
\end{document}), are uniformly bounded in the spaces \documentclass{aastex}
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\begin{document}
$L^p\big(x^{2(\alpha+\gamma)+1}\,dx\big)$
\end{document}. Clearly, the convergence \documentclass{aastex}
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\begin{document}
$R_n^{\alpha}f\to f$
\end{document} 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 \documentclass{aastex}
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\begin{document}
$\alpha$
\end{document} are supported on the interval \documentclass{aastex}
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\begin{document}
$[0,1]$
\end{document}.