Generating functions for B-splines with knots in geometric or affine progression
type: Item
instance of: scholarly article
title: Generating functions for B-splines with knots in geometric or affine progression
zbMATH Open document ID: 1316.65020
author: Çetin Dişibüyük
author: Gülter Budakçı
author: Halil Oruç
author: Ron Goldman
published in: Calcolo
publication date: 2015-03-23T00:00:00Z
review text: The authors obtain generating functions for B-splines. Since there is no structure on the knots when they are completely arbitrary, any interesting property about the corresponding B-splines cannot be derived. Here explicit formulas for the generating functions of B-splines with knots sequences \(\{x_k\}\) in affine progression \(x_{k+1}=qx_k+h\) are given, with \(q\) and \(h\) fixed parameters, and in the particular case (\(h=0\)) of geometric progression they are considered in terms of \(q\)-exponential functions. To find an explicit formula for the generating functions by the de Boor recurrence over knots in affine progression and the definition of \(q\)-derivative, a partial differential equation is constructed, whose solution is the generating function for B-splines in affine progression. Such generating functions are then used to obtain both known and new identities, as a generalization of the Schoenberg identity and an explicit expression for the moments of these B-splines. Finally, this discussion includes two interesting special cases, i.e., uniform B-splines with knots at the integers (\(q=h=1\)) and nonuniform B-splines with knots at the \(q\)-integers (\(q\not=1\), \(h=1\)).
reviewed by: Paola Lamberti
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 6418691
zbMATH Keywords: B-splines
zbMATH Keywords: generating functions
zbMATH Keywords: knots in affine progression
zbMATH Keywords: knots in geometric progression
zbMATH Keywords: \textit{q}-derivatives
zbMATH Keywords: \textit{q}-exponentials
MaRDI profile type: Publication
full work available at URL: s10092-013-0102-8
OpenAlex ID: W2169487856
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DOI: 10.1007/S10092-013-0102-8
description: scientific article; zbMATH DE number 6418691