Symmetric four-directional bivariate pseudo-splines

TL;DR

This paper develops algebraic formulas for symmetric four-directional bivariate pseudo-splines, extending univariate pseudo-spline concepts to two dimensions with minimal support for specific polynomial properties.

Contribution

It provides a formula for constructing the symbols of symmetric four-directional bivariate pseudo-splines, advancing the theory of multivariate subdivision schemes.

Findings

Derived algebraic formulas for bivariate pseudo-spline symbols

Extended univariate pseudo-spline theory to two dimensions

Facilitated construction of minimal support schemes with polynomial properties

Abstract

Univariate pseudo-splines are a generalization of uniform B-splines and interpolatory $2n$-point subdivision schemes. Each pseudo-spline is characterized as the subdivision scheme with least possible support among all schemes with specific degrees of polynomial generation and reproduction. In this paper we consider the problem of constructing the symbols of the bivariate counterpart and provide a formula for the symbols of a family of symmetric four-directional bivariate pseudo-splines. All methods employed are of purely algebraic nature.