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On The Convolution of a Box Spline With a Compactly Supported Distribution: Linear Independence for the Integer Translates

Published online by Cambridge University Press:  20 November 2018

Charles K. Chui  and
Amos Ron
Charles K. Chui
Affiliation:
Department of Mathematics Texas A & M University College Station, Texas 77843 U.S.A.
Amos Ron
Affiliation:
Amos Ron Center for the Mathematical Sciences and Computer Sciences Department University of Wisconsin-Madison 1210 West Dayton St. Madison, Wisconsin 53706 USA.
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Abstract

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The problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

Keywords

box splinesexponential box splinescompactly supported functionsinteger translateslinear independencemultivariate splines41A6341A15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 1 , 01 February 1991 , pp. 19 - 33
Copyright
Copyright © Canadian Mathematical Society 1991

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