Refinement Equations and Spline Functions

Details Refinement Equations and Spline Functions Refinement Equations and Spline Functions

2008-04-14

arxiv.org/abs/0804.2203v1

Mathematics

Numerical Analysis

Scientific paper

In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial {\bf C}^{\infty} solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.

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