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Erschienen in:
2016 | OriginalPaper | Buchkapitel
Localized Summability Kernels for Jacobi Expansions
verfasst von : H. N. Mhaskar
Erschienen in: Mathematical Analysis, Approximation Theory and Their Applications
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Abstract
While the direct and converse theorems of approximation theory enable us to characterize the smoothness of a function \(f: [-1,1] \rightarrow \mathbb{R}\) 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.