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Approximation order and approximate sum rules in subdivision
- Approximation order and approximate sum rules in subdivision
- Conti, Costanza; Romani, Lucia; Yoon, Jungho
- Ewha Authors
- SCOPUS Author ID
- Issue Date
- Journal Title
- JOURNAL OF APPROXIMATION THEORY
- JOURNAL OF APPROXIMATION THEORY vol. 207, pp. 380 - 401
- Subdivision schemes; Exponential polynomial generation and reproduction; Asymptotical similarity; Approximate sum rules; Approximation order
- ACADEMIC PRESS INC ELSEVIER SCIENCE
- SCIE; SCOPUS
- Document Type
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- 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. (C) 2016 Elsevier Inc. All rights reserved.
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