Approximation order and approximate sum rules in subdivision

TL;DR

This paper explores how exponential polynomials and approximate sum rules influence the approximation properties of non-stationary subdivision schemes, extending known stationary scheme results to the non-stationary context.

Contribution

It establishes that exponential polynomials and approximate sum rules play roles analogous to polynomials and sum rules in stationary schemes, with new results on reproduction, generation, and convergence.

Findings

Reproduction of exponential polynomials implies approximate sum rules of corresponding order.

Generation of exponential polynomials implies approximate sum rules under additional conditions.

Reproduction of exponential polynomial spaces and asymptotical similarity ensure approximation order N.

Abstract

Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum rules, its convergence, smoothness and approximation order. The aim of this paper is to show that, in the non-stationary case, exponential polynomials and approximate sum rules play an analogous role of polynomials and sum rules in the stationary case. Indeed, in the non-stationary univariate case we are able to show the following important facts: i) reproduction of $N$ exponential polynomials implies approximate sum rules of order $N$; ii) generation of $N$ exponential polynomials implies approximate sum rules of order $N$, under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; iii) reproduction of an $N$-dimensional space of exponential polynomials and asymptotical similarity imply approximation order $N$; iv) the sequence of basic limit functions of a non-stationary scheme reproducing one exponential polynomial converges uniformly to the basic limit function of the asymptotically similar stationary scheme.