Localized Summability Kernels for Jacobi Expansions

H. N. Mhaskar ()
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H. N. Mhaskar: California Institute of Technology

A chapter in Mathematical Analysis, Approximation Theory and Their Applications, 2016, pp 417-434 from Springer

Abstract: Abstract While the direct and converse theorems of approximation theory enable us to characterize the smoothness of a function f : [ − 1 , 1 ] → ℝ $$f: [-1,1] \rightarrow \mathbb{R}$$ in terms of its degree of polynomial approximation, they do not account for local smoothness. The use of localized summability kernels leads to a wavelet-like representation, using the Fourier–Jacobi coefficients of f, so as to characterize the smoothness of f in a neighborhood of each point in terms of the behavior of the terms of this representation. In this paper, we study the localization properties of a class of kernels, which have explicit forms in the “space domain,” and establish explicit bounds on the Lebesgue constants on the summability kernels corresponding to some of these.

Keywords: Jacobi expansions; Localized kernels; Jacobi translates; Product formulas (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-31281-1_18

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DOI: 10.1007/978-3-319-31281-1_18

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