On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. III
type: Item
instance of: scholarly article
title: On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. III
zbMATH Open document ID: 0926.65001
DOI: 10.1023/A:1022317224199
published in: BIT
publication date: 1999-11-30T00:00:00Z
review text: This paper is the third and last one in a series of publications concerned with several summation formulas [for part II see ibid. 37, No. 4, 804-832 (1997; Zbl 0893.65001)]. The main emphasis is now on Planas summation formula, which is \[ \sum^\infty_{k= m} f(k)= {f(m)\over 2}+ \int^\infty_m f(\sigma) d\sigma- \int^\infty_{-\infty} g^P_m(\omega) w^P(\omega) d\omega, \] \[ \lim_{n\to\infty} T(m:n)f= -\int^\infty_{-\infty} g^P_m(\omega) w^P(\omega) d\omega. \] Here, \[ g^P_\sigma(\omega)= {f(\sigma+ i\omega)- f(\sigma- i\omega)\over 2i\omega},\quad w^P(\omega)= {|\omega|\over e^{| 2\pi\omega|}- 1} \] and \(f\) has to satisfy several technical assumptions. Moreover, some related density functions and summation formulas are investigated. Throughout the whole paper, a lot of numerical examples illustrate the author's considerations.
reviewed by: Guido Walz
Mathematics Subject Classification ID: 65B10
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65E05
zbMATH DE Number: 1275242
zbMATH Keywords: Plana summation
zbMATH Keywords: Gauss-Christoffel rule
zbMATH Keywords: summation formulas
zbMATH Keywords: numerical examples
author: Germund Dahlquist
MaRDI profile type: Publication
full work available at URL: a:1022317224199
OpenAlex ID: W4247920008
description: scientific article; zbMATH DE number 1275242