Wiener type theorems for Jacobi series with nonnegative coefficients

Mhaskar, H.; Tikhonov, S.

doi:10.1090/S0002-9939-2011-10964-X

Wiener type theorems for Jacobi series with nonnegative coefficients
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by H. N. Mhaskar and S. Tikhonov PDF

Proc. Amer. Math. Soc. 140 (2012), 977-986 Request permission

Abstract:

This paper gives three theorems regarding functions integrable on $[-1,1]$ with respect to Jacobi weights and having nonnegative coefficients in their (Fourier–)Jacobi expansions. We show that the $L^p$-integrability (with respect to the Jacobi weight) on an interval near $1$ implies the $L^p$-integrability on the whole interval if $p$ is an even integer. The Jacobi expansion of a function locally in $L^\infty$ near $1$ is shown to converge uniformly and absolutely on $[-1,1]$; in particular, such a function is shown to be continuous on $[-1,1]$. Similar results are obtained for functions in local Besov approximation spaces.

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