On tensor product approximation of analytic functions
On tensor product approximation of analytic functions
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2015Verlag / herausgebende Einrichtung
Institut für Numerische Simulation (INS)Verlagsort
BonnMetadaten
Zur Langanzeige
Zitierbarer Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11890
Inhalt
We prove sharp, two-sided bounds on sums of the form ∑k∈ℕd0\𝒟a(T) exp(− ∑dj=1 ajkj), where 𝒟a(T) := {k ∈ ℕd0 : ∑dj=1 ajkj ≤ T} and a ∈ ℝd+. 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 [6] or interpolation and integration rules at Leja points [13, 40, 41]. Moreover, we discuss the limit d → ∞, 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.
Schlagwörter
Multivariate approximation, Interpolation, Integration, Sparse grids, Infinite dimensions, Analytic functions
Klassifikation (DDC)
510 Mathematik 518 Numerische AnalysisErstpublikation
https://ins.uni-bonn.de/publication/preprintsZugehörige Publikation(en)
https://doi.org/10.1016/j.jat.2016.02.006Reihe, Nummer
INS Preprints ; 1512Griebel, Michael; Oettershagen, Jens: On tensor product approximation of analytic functions. In: INS Preprints, 1512.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11890
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11890
@unpublished{handle:20.500.11811/11890,
author = {{Michael Griebel} and {Jens Oettershagen}},
title = {On tensor product approximation of analytic functions},
publisher = {Institut für Numerische Simulation (INS)},
year = 2015,
INS Preprints},
volume = 1512,
note = {We prove sharp, two-sided bounds on sums of the form ∑k∈ℕd0\𝒟a(T) exp(− ∑dj=1 ajkj), where 𝒟a(T) := {k ∈ ℕd0 : ∑dj=1 ajkj ≤ T} and a ∈ ℝd+. 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 [6] or interpolation and integration rules at Leja points [13, 40, 41]. Moreover, we discuss the limit d → ∞, 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.},
url = {https://hdl.handle.net/20.500.11811/11890}
}
author = {{Michael Griebel} and {Jens Oettershagen}},
title = {On tensor product approximation of analytic functions},
publisher = {Institut für Numerische Simulation (INS)},
year = 2015,
INS Preprints},
volume = 1512,
note = {We prove sharp, two-sided bounds on sums of the form ∑k∈ℕd0\𝒟a(T) exp(− ∑dj=1 ajkj), where 𝒟a(T) := {k ∈ ℕd0 : ∑dj=1 ajkj ≤ T} and a ∈ ℝd+. 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 [6] or interpolation and integration rules at Leja points [13, 40, 41]. Moreover, we discuss the limit d → ∞, 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.},
url = {https://hdl.handle.net/20.500.11811/11890}
}
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