Approximation order and approximate sum rules in subdivision
For researchers in subdivision schemes, this work extends key theoretical connections from stationary to non-stationary cases, providing foundational understanding.
The paper establishes that in non-stationary subdivision schemes, exponential polynomial reproduction and generation imply approximate sum rules of the same order, and together with asymptotical similarity yield approximation order N. It also shows convergence of basic limit functions to those of the stationary counterpart.
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.