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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eigenvalues of scaling operators and a characterization of $B$-splines


Authors: Xiaojie Gao, S. L. Lee and Qiyu Sun
Journal: Proc. Amer. Math. Soc. 134 (2006), 1051-1057
MSC (2000): Primary 41A15, 41A99, 42C40, 65T60
Published electronically: July 21, 2005
MathSciNet review: 2196038
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Abstract: A finitely supported sequence $a$ that sums to $2$ defines a scaling operator $ T_a f = \sum_{k\in \mathbb Z} a(k)f(2 \cdot -k)$ on functions $f,$ a transition operator $S_a v = \sum_{k\in \mathbb Z} a(k) (2 \cdot -k)$ on sequences $v,$ and a unique compactly supported scaling function $\phi$ that satisfies $\phi = T_a \phi$normalized with $\widehat \phi (0) = 1.$ It is shown that the eigenvalues of $T_a$ on the space of compactly supported square-integrable functions are a subset of the nonzero eigenvalues of the transition operator $S_a$ on the space of finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function $\phi$is a uniform $B$-spline.


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Additional Information

Xiaojie Gao
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matgxj@nus.edu.sg

S. L. Lee
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matleesl@nus.edu.sg

Qiyu Sun
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matsungy@nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08092-5
PII: S 0002-9939(05)08092-5
Keywords: Scaling operators, transition operators, eigenvalues, uniform $B$-splines
Received by editor(s): October 25, 2004
Published electronically: July 21, 2005
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society