Refinement Equations and Spline Functions
Numerical Analysis
2008-04-15 v1 Number Theory
Abstract
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.
Cite
@article{arxiv.0804.2203,
title = {Refinement Equations and Spline Functions},
author = {Artūras Dubickas and Zhiqiang Xu},
journal= {arXiv preprint arXiv:0804.2203},
year = {2008}
}