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

 

Linear independence of pseudo-splines


Authors: Bin Dong and Zuowei Shen
Journal: Proc. Amer. Math. Soc. 134 (2006), 2685-2694
MSC (2000): Primary 42C40, 41A30
Published electronically: March 23, 2006
MathSciNet review: 2213748
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Abstract: In this paper, we show that the shifts of a pseudo-spline are linearly independent. This is stronger than the (more obvious) statement that the shifts of a pseudo-spline form a Riesz system. In fact, the linear independence of a compactly supported (refinable) function and its shifts has been studied in several areas of approximation and wavelet theory. Furthermore, the linear independence of the shifts of a pseudo-spline is a necessary and sufficient condition for the existence of a compactly supported function whose shifts form a biorthogonal dual system of the shifts of the pseudo-spline.


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Additional Information

Bin Dong
Affiliation: Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543, Singapore
Address at time of publication: Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, California 90095-1555
Email: g0301173@nus.edu.sg; bdong@math.ucla.edu

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543, Singapore
Email: matzuows@nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08316-X
PII: S 0002-9939(06)08316-X
Keywords: Linear independence, pseudo-spline, stability.
Received by editor(s): September 22, 2004
Received by editor(s) in revised form: April 6, 2005
Published electronically: March 23, 2006
Additional Notes: This research was supported by several grants from the Department of Mathematics, National University of Singapore.
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.