Gauss-Hermite quadratures for functions from Hilbert spaces with Gaussian reproducing kernels
type: Item
instance of: scholarly article
title: Gauss-Hermite quadratures for functions from Hilbert spaces with Gaussian reproducing kernels
zbMATH Open document ID: 1247.65034
author: H. Woźniakowski
author: F. Y. Kuo
published in: BIT
publication date: 2012-07-31T00:00:00Z
review text: Let \(H(K_{\gamma})\) be the Hilbert space with the Gaussian reproducing kernel \(K_{\gamma}=\exp(-\gamma^2(x-y)^2)\) for \(x,\,y\in \mathbb R\) and \(\gamma >0\). For \(f\in H(K_{\gamma})\), the authors study Gauss-Hermite quadratures for the integral \[ I(f) = \int_{-\infty}^{\infty} f(x)\, \exp(-x^2/(2\alpha))\,dx \] with \(\alpha>0\). The Gauss-Hermite quadratures approximate \(I(f)\) by \[ Q_n(f) = \sum_{j=1}^n w_j\, f(x_j)\,, \] where \(w_j\) are positive weights and \(x_j\) are the zeros of the generalized Hermite polynomial \[ H_n^{[\alpha]}(x) = (2\alpha)^{-n/2}\, H_n(x\,(2\alpha)^{-1/2})\,. \] Here \(H_n\) denotes the standard Hermite polynomial. The authors prove an explicit formula for the error of Gauss-Hermite quadrature. Thus, for \(2\alpha \gamma^2 < 1\) one has an exponential convergence rate of the Gauss-Hermite quadratures as \(n\to \infty\). In the case \(2\alpha \gamma^2 = 1\) one has no convergence. For \(2\alpha \gamma^2 > 1\) one has an exponential rate of divergence.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 6062472
zbMATH Keywords: Gauss-Hermite quadrature
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: explicit error formula
zbMATH Keywords: exponential rate of convergence
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Gaussian kernel
zbMATH Keywords: integration with Gaussian weight
describes a project that uses: ElemStatLearn
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: s10543-011-0358-9
OpenAlex ID: W1990166848
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DOI: 10.1007/S10543-011-0358-9
description: scientific article; zbMATH DE number 6062472