Bandlimited approximations to the truncated Gaussian and applications
type: Item
instance of: scholarly article
title: Bandlimited approximations to the truncated Gaussian and applications
zbMATH Open document ID: 1280.41018
author: Emanuel Carneiro
author: Friedrich Littmann
published in: Constructive Approximation
publication date: 2013-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.0567
review text: The authors find the explicit solutions of the best \( L^1(\mathbb{R}) \) (unrestricted and one-sided) approximation problems by entire functions of exponential type of at most \( \pi \) of the truncated and the odd Gaussians \[ G_{\lambda}^+(x)=x_+^0e^{-\pi\lambda x^2},\quad G_{\lambda}^o(x)=\mathrm{sign}(x)e^{-\pi\lambda x^2}. \] For example, the best \( L^1(\mathbb{R}) \) approximation of \(G_{\lambda}^+ \) is \[ K_{\lambda}^+(z)=\frac{\sin\pi z}{\pi}\sum_{n=1}^{\infty}(-1)^n\left\{\frac{G_{\lambda}^+(n)}{z-n}-\frac{G_{\lambda}^+(n)}{n}\right\}. \] The second part of the paper is devoted to the integration on the free parameter \( \lambda \) as a tool to generate the solution of these best approximation problems for a class of truncated and odd functions.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 42A82
Mathematics Subject Classification ID: 30D10
Mathematics Subject Classification ID: 41A29
zbMATH DE Number: 6197820
zbMATH Keywords: truncated Gaussian
zbMATH Keywords: exponential type
zbMATH Keywords: extremal functions
zbMATH Keywords: one-sided best approximation
zbMATH Keywords: tempered distributions
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
OpenAlex ID: W2055333241
arXiv ID: 1106.0567
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DOI: 10.1007/S00365-012-9177-8
description: scientific article; zbMATH DE number 6197820