Impossibility of fast stable approximation of analytic functions from equispaced samples

Rodrigo Platte; Lloyd N. Trefethen; Arno B J Kuijlaars

doi:10.1137/090774707

Impossibility of fast stable approximation of analytic functions from equispaced samples

Rodrigo Platte, Lloyd N. Trefethen, Arno B J Kuijlaars

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

105 Scopus citations

Abstract

It is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions. To avoid instability, one must settle for root-exponential convergence. The proof combines a Bernstein inequality of 1912 with an estimate due to Coppersmith and Rivlin in 1992.

Original language	English (US)
Pages (from-to)	308-318
Number of pages	11
Journal	SIAM Review
Volume	53
Issue number	2
DOIs	https://doi.org/10.1137/090774707
State	Published - 2011

Keywords

Gibbs phenomenon
Interpolation
Lanczos iteration
Radial basis functions
Runge phenomenon

ASJC Scopus subject areas

Theoretical Computer Science
Computational Mathematics
Applied Mathematics

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10.1137/090774707

Cite this

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title = "Impossibility of fast stable approximation of analytic functions from equispaced samples",

abstract = "It is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions. To avoid instability, one must settle for root-exponential convergence. The proof combines a Bernstein inequality of 1912 with an estimate due to Coppersmith and Rivlin in 1992.",

keywords = "Gibbs phenomenon, Interpolation, Lanczos iteration, Radial basis functions, Runge phenomenon",

author = "Rodrigo Platte and Trefethen, {Lloyd N.} and Kuijlaars, {Arno B J}",

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doi = "10.1137/090774707",

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N2 - It is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions. To avoid instability, one must settle for root-exponential convergence. The proof combines a Bernstein inequality of 1912 with an estimate due to Coppersmith and Rivlin in 1992.

AB - It is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions. To avoid instability, one must settle for root-exponential convergence. The proof combines a Bernstein inequality of 1912 with an estimate due to Coppersmith and Rivlin in 1992.

KW - Gibbs phenomenon

KW - Interpolation

KW - Lanczos iteration

KW - Radial basis functions

KW - Runge phenomenon

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Impossibility of fast stable approximation of analytic functions from equispaced samples

Abstract

Keywords

ASJC Scopus subject areas

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