Analysis of optimal bivariate symmetric refinable Hermite interpolants

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 shall 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 shall 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 shall 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 $[-1,1]^2$.