Please use this identifier to cite or link to this item: https://doi.org/10.1090/S0002-9939-05-08092-5
Title: Eigenvalues of scaling operators and a characterization of B-splines
Authors: GAO XIAOJIE 
Lee, S.L. 
Sun, Q. 
Keywords: Eigenvalues
Scaling operators
Transition operators
Uniform B-splines
Issue Date: Apr-2006
Citation: GAO XIAOJIE, Lee, S.L., Sun, Q. (2006-04). Eigenvalues of scaling operators and a characterization of B-splines. Proceedings of the American Mathematical Society 134 (4) : 1051-1057. ScholarBank@NUS Repository. https://doi.org/10.1090/S0002-9939-05-08092-5
Abstract: A finitely supported sequence a that sums to 2 defines a scaling operator Taf = ∑k∈z a(k) f (2·- k) on functions f, a transition operator Sav = ∑k∈z a(k) (2·- k) on sequences v, and a unique compactly supported scaling function φ that satisfies φ = Taφ normalized with φ̂(0) = 1. It is shown that the eigenvalues of Ta on the space of compactly supported square-integrable functions are a subset of the nonzero eigenvalues of the transition operator Sas on the space of finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function φ is a uniform B-spline. ©2005 American Mathematical Society.
Source Title: Proceedings of the American Mathematical Society
URI: http://scholarbank.nus.edu.sg/handle/10635/103192
ISSN: 00029939
DOI: 10.1090/S0002-9939-05-08092-5
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