2008/04/14 by Artūras Dubickas, Dubickas, Artūras, Zhiqiang Xu +1
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Digital Filter Design and Implementation #FOS: Mathematics #Number Theory (math.NT) #Numerical Analysis (math.NA) #Numerical Methods and Algorithms
paper · pdf · doi:10.48550/arxiv.0804.2203
openalex publication_date 2008/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∞ 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.