On tensor product approximation of analytic functions

JOURNAL OF APPROXIMATION THEORY

卷:207

Article

Griebel, Michael^1,2  Oettershagen, Jens¹ 

[1] Univ Bonn, Inst Numer Simulat, Wegelerstr 6, D-53115 Bonn, Germany

[2] Fraunhofer Inst Algorithms & Sci Comp, Bonn, Germany

关键词: Multivariate approximation;    Interpolation;    Integration;    Sparse grids;    Infinite dimensions;    Analytic functions;   

DOI  :  10.1016/j.jat.2016.02.006

来源: Elsevier

【 摘 要 】

We prove sharp, two-sided bounds on sums of the form Sigma(d)(exp)(k epsilon N0)(\Da(T))(-Sigma(d)(j=1) a(j)k(j)), where Da(T) := {k epsilon N-0(d) : Sigma(d)(j=1) a(j)k(j) <= T} and a epsilon R-+(d). These sums appear in the error analysis of tensor product approximation, interpolation and integration of d-variate analytic functions. Examples are tensor products of univariate Fourier-Legendre expansions (Beck et al., 2014) or interpolation and integration rules at Leja points (Chkifa et al., 2013), (Narayan and Jakeman, 2014), (Nobile et al., 2014). Moreover, we discuss the limit d -> infinity, where we prove both, algebraic and sub-exponential upper bounds. As an application we consider tensor products of Hardy spaces, where we study convergence rates of a certain truncated Taylor series, as well as of interpolation and integration using Leja points. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free