TY - JOUR

T1 - Approximation order and approximate sum rules in subdivision

AU - Conti, Costanza

AU - Romani, Lucia

AU - Yoon, Jungho

N1 - Funding Information:
Support from the Italian GNCS-INdAM within the research project entitled “Theoretical advances, computational methods and new applications for interpolants and approximants in spaces of generalized splines” is gratefully acknowledged. Lucia Romani acknowledges the support received from Ministero dell’Istruzione, dell’Università e della Ricerca —Progetti di Ricerca di Interesse Nazionale 2012 (MIUR-PRIN 2012—grant 2012MTE38N ). Jungho Yoon acknowledges the support received from the National Research Foundation of Korea through the grants NRF-2015-R1A5A1009350 (Science Research Center Program) and NRF-2015-R1D1A1A09057553 .
Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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.

AB - 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.

KW - Approximate sum rules

KW - Approximation order

KW - Asymptotical similarity

KW - Exponential polynomial generation and reproduction

KW - Subdivision schemes

UR - http://www.scopus.com/inward/record.url?scp=84962835205&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2016.02.014

DO - 10.1016/j.jat.2016.02.014

M3 - Article

AN - SCOPUS:84962835205

SN - 0021-9045

VL - 207

SP - 380

EP - 401

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

ER -