Approximation order and approximate sum rules in subdivision

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• 作者：Costanza Conti^a ; ^{costanza.conti@unifi.it" class="auth_mail" title="E-mail the corresponding author} ; Lucia Romani^b ; ^{lucia.romani@unimib.it" class="auth_mail" title="E-mail the corresponding author} ; Jungho Yoon^c ; ^{yoon@ewha.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65D17 ; 65D15 ; 41A25 ; 41A10 ; 41A30
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：207
• 期：Complete
• 页码：380-401
• 全文大小：296 K

文摘

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$N$ exponential polynomials implies approximate sum rules of order N$N$; (ii) generation of N$N$ exponential polynomials implies approximate sum rules of order N$N$, under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; (iii) reproduction of an N$N$-dimensional space of exponential polynomials and asymptotical similarity imply approximation order N$N$; (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.