Refinement equations and spline functions

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 C ∞ 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.

Matrix refinement type equations

“…[6,8]), spline theory (see e.g. [7,11]), subdivision schemes in approximation theory and curve design (see e.g. [2,21]), combinatorial number theory (see [24]).…”

Inhomogeneous poly-scale refinement type equations and Markov operators with perturbations

“…It turns out that in some applications, continuous and bounded or continuous and compactly supported solutions of homogeneous refinement equations are important. Such solutions have significant applications in wavelet theory, approximation theory, theory of subdivision schemes, computer graphics, physics, combinatorial number theory and many others (see, e.g., [2,7,8,9,18,19,22]). From the point of view of applications, all results on the existence of "good" solutions of homogeneous as well as of inhomogeneous refinement equations are very important (see [3] where it is showed how nonexistence of "good" solutions of a refinement equation can lead to anomalous behavior of numerical methods for a construction of wavelets).…”

The regularity of refinable functions

“…This is perhaps due to the fact that the matrix technique that has been effective for the integral case can no longer be applied. Using techniques from number theory and harmonic analysis, Dubickas and Xu [8] prove that a refinable function in R with an arbitrary dilation λ and integer translations cannot be in C ∞ . This appears to be the only generalization in this direction.…”