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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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202110209 |
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| _version_ | 1850235307726733312 |
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| author | Janusz Morawiec |
| author_facet | Janusz Morawiec |
| author_sort | Janusz Morawiec |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-104a3938c36d4997a5000f339883bc8b |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-104a3938c36d4997a5000f339883bc8b2025-08-20T02:02:19ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0132313914810.1155/S0161171202110209On local properties of compactly supported solutions of the two-coefficient dilation equationJanusz Morawiec0Instytut Matematyki, Uniwersytet Śląski, ul. Bankowa 14, Katowice PL-40-007, PolandLet 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.http://dx.doi.org/10.1155/S0161171202110209 |
| spellingShingle | Janusz Morawiec On local properties of compactly supported solutions of the two-coefficient dilation equation International Journal of Mathematics and Mathematical Sciences |
| title | On local properties of compactly supported solutions of the two-coefficient dilation equation |
| title_full | On local properties of compactly supported solutions of the two-coefficient dilation equation |
| title_fullStr | On local properties of compactly supported solutions of the two-coefficient dilation equation |
| title_full_unstemmed | On local properties of compactly supported solutions of the two-coefficient dilation equation |
| title_short | On local properties of compactly supported solutions of the two-coefficient dilation equation |
| title_sort | on local properties of compactly supported solutions of the two coefficient dilation equation |
| url | http://dx.doi.org/10.1155/S0161171202110209 |
| work_keys_str_mv | AT januszmorawiec onlocalpropertiesofcompactlysupportedsolutionsofthetwocoefficientdilationequation |