Refinement Equations and Spline Functions

TL;DR

This paper investigates the regularity of refinable functions with non-integer dilations, proving nonexistence of smooth solutions, extending spline results, and analyzing box splines using number theory and harmonic analysis.

Contribution

It establishes the nonexistence of smooth solutions for refinement equations with non-integer dilations and extends the theory of refinable splines, providing new counterexamples and analysis.

Findings

No non-trivial C^{} solutions exist for certain refinement equations.

Extended results on refinable splines with non-integer dilations.

Constructed counterexamples to existing conjectures.

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.