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Local Approximation Using Hermite Functions

Mhaskar, H. N. (2017) Local Approximation Using Hermite Functions. In: Progress in Approximation Theory and Applicable Complex Analysis: In Memory of Q.I. Rahman. Springer Optimization and Its Applications. No.117. Springer , Cham, Switzerland, pp. 341-362. ISBN 978-3-319-49240-7.

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We develop a wavelet-like representation of functions in Lp(R) based on their Fourier–Hermite coefficients; i.e., we describe an expansion of such functions where the local behavior of the terms characterize completely the local smoothness of the target function. In the case of continuous functions, a similar expansion is given based on the values of the functions at arbitrary points on the real line. In the process, we give new proofs for the localization of certain kernels, as well as for some very classical estimates such as the Markov–Bernstein inequality.

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Additional Information:© 2017 Springer International Publishing AG. First Online: 04 April 2017. The research of HNM is supported in part by Grant W911NF-15-1-0385 from the US Army Research Office. We thank the editors for their kind invitation to submit this paper.
Funding AgencyGrant Number
Army Research Office (ARO)W911NF-15-1-0385
Subject Keywords:Approximation with Hermite polynomials; Localized kernels; Quadrature formulas; Wavelet-like representation
Series Name:Springer Optimization and Its Applications
Issue or Number:117
Record Number:CaltechAUTHORS:20170711-111543165
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78946
Deposited By: Tony Diaz
Deposited On:11 Jul 2017 21:12
Last Modified:03 Oct 2019 18:14

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