International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 3, Pages 139-148
doi:10.1155/S0161171202110209
On local properties of compactly supported solutions of the two-coefficient dilation equation
Instytut Matematyki, Uniwersytet Śląski, ul. Bankowa 14, Katowice PL-40-007, Poland
Received 22 October 2001
Copyright © 2002 Janusz Morawiec. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let a and b be reals. We consider the compactly supported solutions φ:ℝ→ℝ of the two-coefficient dilation equation
φ(x)=aφ(2x)+bφ(2x−1). In this paper, we determine sets Ba,b, Ca,b, and Za,b defined in the following way: let x∈[0,1]. We say that x∈Ba,b (resp., x∈Ca,b, x∈Za,b) if the zero function is the only compactly supported solution of the two-coefficient dilation equation, which is bounded in a neighbourhood of x (resp., continuous at x, vanishes in a neighbourhood of x). We also give the structure of the general compactly supported solution of the two-coefficient dilation equation.