Polynomial Reproduction of Multivariate Scalar Subdivision Schemes with General Dilation

TL;DR

This paper establishes algebraic conditions for polynomial reproduction in multivariate scalar subdivision schemes with general dilation matrices, enabling improved scheme design and understanding of approximation properties.

Contribution

It provides algebraic criteria for polynomial reproduction, allowing the enhancement and construction of subdivision schemes with desired polynomial reproduction degrees.

Findings

Derived algebraic conditions for polynomial reproduction

Determined approximation order of associated functions

Constructed schemes with specified polynomial reproduction degree

Abstract

In this paper we study scalar multivariate subdivision schemes with general integer expanding dilation matrix. Our main result yields simple algebraic conditions on the symbols of such schemes that characterize their polynomial reproduction, i.e. their capability to generate exactly the same polynomials from which the initial data is sampled. These algebraic conditions also allow us to determine the approximation order of the associated refinable functions and to choose the "correct" parametrization, i.e. the grid points to which the newly computed values are attached at each subdivision iteration. We use this special choice of the parametrization to increase the degree of polynomial reproduction of known subdivision schemes and to construct new schemes with given degree of polynomial reproduction.