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Local interpolation with optimal polynomial exactness in refinement spaces


Authors: Johan de Villiers and Mpfareleni Rejoyce Gavhi
Journal: Math. Comp. 85 (2016), 759-782
MSC (2010): Primary 65D05; Secondary 65D07
DOI: https://doi.org/10.1090/mcom/3006
Published electronically: June 25, 2015
MathSciNet review: 3434880
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Abstract: A constructive existence result for a class of polynomial identities is established and applied in two related contexts. First, an algorithm is developed for the explicit construction of a sequence of local interpolation operators mapping the space of continuous functions on the real line into the nested sequence of refinement spaces generated by the shifts of a given refinable function, and where optimal polynomial exactness, as governed by the order of the sum-rule condition satisfied by the corresponding refinement sequence, is achieved. The above algorithm requires as input only the values at the integers of the refinable function, and we proceed, secondly, to derive sufficient conditions for the existence of a refinable function with prescribed values at the integers. As our main examples, we consider the cardinal $ B$-spline case, as well as refinable functions with normalized binomial coefficient values at the integers.


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Additional Information

Johan de Villiers
Affiliation: Department of Mathematical Sciences, Mathematics Division, Stellenbosch University, South Africa — and — African Institute for Mathematical Sciences (AIMS), Muizenberg, South Africa

Mpfareleni Rejoyce Gavhi
Affiliation: Department of Mathematical Sciences, Mathematics Division, Stellenbosch University, South Africa — and — African Institute for Mathematical Sciences (AIMS), Muizenberg, South Africa

DOI: https://doi.org/10.1090/mcom/3006
Received by editor(s): May 17, 2013
Received by editor(s) in revised form: August 22, 2014
Published electronically: June 25, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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