On local properties of compactly supported solutions of the two-coefficient dilation equation
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...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202110209 |
| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |