Polynomial Reproduction of Multivariate Scalar Subdivision Schemes with General Dilation
For researchers in subdivision schemes and approximation theory, it offers a theoretical framework to analyze and design schemes with desired polynomial reproduction properties.
The paper provides algebraic conditions on the symbols of multivariate scalar subdivision schemes with general dilation that characterize polynomial reproduction, enabling determination of approximation order and optimal parametrization to increase reproduction degree.
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.