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.
- Approximate sum rules
- Approximation order
- Asymptotical similarity
- Exponential polynomial generation and reproduction
- Subdivision schemes