The Polynomials in the Linear Span of Integer Translates of a Compactly Supported Function.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The linear span of integer translates of a fixed compactly supported function phi provides a particularly simple model of an approximating family of the finite element type. The approximating power of such a span or, more precisely, of its scaled versions has been known for some time to be characterizable in terms of the space pi phi of polynomials it contains. Recent work on box splines has provided concrete examples of interest in a multivariate settings and so rekindled interest in the space pi phi. The report derives and extends specific information about pi phi contained in recent work by Dahmen and Micchelli, and by Chui, Diamond, Jetter, Lai and Ward, but does so without reference to specific properties such as piecewise polynomiality, or factorizablity of the Fourier transform of phi. Understanding, in the simplest possible and most efficient terms, of the approximation power of such spaces may provide the necessary insight into approximation by smooth piecewise polynomials on regular, and perhaps even not so regular, partitions. Keywords Quasi-interpolants Invariance and Box splines.
- Theoretical Mathematics