Norm-preserving L−L integral transformations
In this paper we consider an L−L integral transformation G of the form F(x)=∫0∞G(x,y)f(y)dy, where G(x,y) is defined on D={(x,y):x≥0,y≥0} and f(y) is defined on [0,∞). The following results are proved: For an L−L integral transformation G to be norm-preserving, ∫0∞|G*(x,t)|dx=1 for almost all t≥0 is...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1985-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171285000485 |
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| Summary: | In this paper we consider an L−L integral transformation G of the form F(x)=∫0∞G(x,y)f(y)dy, where G(x,y) is defined on D={(x,y):x≥0,y≥0} and f(y) is defined on [0,∞). The following results are proved: For an L−L integral transformation G to be norm-preserving, ∫0∞|G*(x,t)|dx=1 for almost all t≥0 is only a necessary condition, where G*(x,t)=limh→0inf1h∫tt+hG(x,y)dy for each x≥0. For certain G's. ∫0∞|G*(x,t)|dx=1 for almost all t≥0 is a necessary and sufficient condition for preserving the norm of certain f ϵ L. In this paper the analogous result for sum-preserving L−L integral transformation G is proved. |
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| ISSN: | 0161-1712 1687-0425 |