Some qualitative properties of bivariate Euler-Frobenius polynomials
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 \(P_ n\).
multivariate Euler-Frobenius polynomials
scientific article; zbMATH DE number 4034289