Constructive Approximation

, Volume 5, Issue 1, pp 297–308

A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution

  • Amos Ron

DOI: 10.1007/BF01889611

Cite this article as:
Ron, A. Constr. Approx (1989) 5: 297. doi:10.1007/BF01889611


Given a multivariate compactly supported distributionϕ, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of\(\hat \varphi\)=the Fourier-Laplace transform ofϕ. The utility of the condition is demonstrated by several examples and applications, showing, in particular, that previous results on box splines and exponential box splines can be derived from this condition by a simple combinatorial argument.

AMS classification

41A63 41A15 

Key words and phrases

Box splines Exponential box splines Polynomial box splines Integer translates Compactly supported function Compactly supported distribution Spectral analysis Global linear independence Fourier transform 

Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • Amos Ron
    • 1
  1. 1.Raymond and Beverley Sackler Faculty of Exact Sciences School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Computer Sciences DepartmentUniversity of Wisconsin-MadisonMadisonUSA

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