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Analysis of optimal bivariate symmetric refinable Hermite interpolants. (English) Zbl 1213.42140

Summary: Multivariate refinable Hermite interpolants with high smoothness and small support are of interest in CAGD and numerical algorithms. In this article, we are particularly interested in analyzing some univariate and bivariate symmetric refinable Hermite interpolants, which have some desirable properties such as short support, optimal smoothness and spline property. We study the projection method for multivariate refinable function vectors and discuss some properties of multivariate spline refinable function vectors. Here, a compactly supported multivariate spline function on \(\mathbb R^s\) just means a function of piecewise polynomials supporting on a finite number of polygonal partition subdomains of \(\mathbb R^s\). We discuss spline refinable function vectors by investigating the structure of the eigenvalues and eigenvectors of the transition operator. To illustrate the results in this paper, we analyze the optimal smoothness and spline properties of some univariate and bivariate refinable Hermite interpolants. For the regular triangular mesh, we obtain a bivariate \(C^2\) symmetric dyadic refinable Hermite interpolant of order 2 whose mask is supported inside \([-1,1]^2\).

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A05 Interpolation in approximation theory
41A15 Spline approximation
41A63 Multidimensional problems
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