On local properties of compactly supported solutions of the two-coefficient dilation equation

Janusz Morawiec

International Journal of Mathematics and Mathematical Sciences, 2002, vol. 32, 1-10

Abstract:

Let a and b be reals. We consider the compactly supported solutions φ : ℝ → ℝ of the two-coefficient dilation equation φ ( x ) = a φ ( 2 x ) + b φ ( 2 x − 1 ) . In this paper, we determine sets B a , b , C a , b , and Z a , b defined in the following way: let x ∈ [ 0 , 1 ] . We say that x ∈ B a , b (resp., x ∈ C a , b , x ∈ Z a , 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.

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:874892

DOI: 10.1155/S0161171202110209

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