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On linear independence for integer translates of a finite number of functions

Published online by Cambridge University Press:  20 January 2009

Rong-Qing Jia
Affiliation:
Department of MathematicsUniversity of AlbertaEdmonton, T6G 2G1, Canada
Charles A. Micchelli
Affiliation:
IBM Research DivisionT. J. Watson Research CenterMathematical Sciences Department Yorktown Heights, NY 10598, USA
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Abstract

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We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

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