Linear independence of pseudo-splines

Dong, Bin; Shen, Zuowei

doi:10.1090/S0002-9939-06-08316-X

Linear independence of pseudo-splines
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by Bin Dong and Zuowei Shen

Proc. Amer. Math. Soc. 134 (2006), 2685-2694

DOI: https://doi.org/10.1090/S0002-9939-06-08316-X

Published electronically: March 23, 2006

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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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