Some qualitative properties of bivariate Euler-Frobenius polynomials
type: Item
instance of: scholarly article
title: Some qualitative properties of bivariate Euler-Frobenius polynomials
zbMATH Open document ID: 0635.41012
DOI: 10.1016/0021-9045(87)90061-X
author: Klaus Höllig
author: Carl De Boor
author: Sherman D. Riemenschneider
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: The paper generalizes the results of Schoenberg related to the theory of univariate cardinal splines. The authors obtain in this note the bivariate analog of this result for box-splines. For a set of vectors \(\Xi =\{\xi_ 1,...,\xi_ n\}\) with \(\xi_{\nu}\in {\mathbb{Z}}^ m\), the box-spline \(M_{\Xi}\) is the functional on \(C_ 0({\mathbb{R}}^ m)\) defined by \(M_{\Xi}\Phi =\int_{[-1/2,1/2]^ n}\Phi (\sum^{n}_{\nu =1}\lambda_{\nu}\xi_{\nu})d\lambda.\) The multivariate Euler-Frobenius polynomials are defined by \(P_{\Xi}(x)=\sum_{j\in {\mathbb{Z}}^ m}M_{\Xi}(j)e^{ijx}\). The authors conjecture that in the symmetric case, when each of the vectors in \(\Xi\) occurs with multiplicity n, the polynomial \(P_ n=P_{\Xi}\) attains its minimum at the point (2\(\pi\) /3, \(2\pi\) /3). They prove here the conjecture and determine all critical points of \(P_ n\). Namely: Theorem: The polynomials \(P_ n\), \(n\in {\mathbb{Z}}_+\), attain their minima at \(\pm (2\pi /3,2\pi /3)mod 2\pi {\mathbb{Z}}^ 2\), their maxima at the points \(2\pi {\mathbb{Z}}^ 2\) and have saddle points at \(\pi {\mathbb{Z}}^ 2mod 2\pi {\mathbb{Z}}^ 2\). These are the only critical points of \(P_ n\).
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65D07
zbMATH DE Number: 4034289
zbMATH Keywords: univariate cardinal splines
zbMATH Keywords: box-spline
zbMATH Keywords: multivariate Euler-Frobenius polynomials
MaRDI profile type: Publication
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full work available at URL: 0021-9045(87)90061-x
OpenAlex ID: W2013788125
description: scientific article; zbMATH DE number 4034289