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Some Results on the Composition of Singular Distributions |
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PP: 623-634 |
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Author(s) |
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Adem Kılıc¸man,
Brian Fisher,
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Abstract |
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| Let F be a distribution in D0 and let f be a locally summable function. The neutrix
composition F(f(x)) of F and f is said to exist and equal to the distribution
h(x) if the neutrix limit of the sequence fFn(f(x))g is equal to h(x), where
Fn(x) = F(x) ¤ ±n(x) for n = 1; 2; : : : and f±n(x)g is a certain regular sequence
converging to the Dirac delta function. In particular, the composition F(f(x)) is said
to exist and be equal to the distribution h if the sequence fFn(f(x))g converges to h
in the normal sense.
In this study it was proved that if F(x) denotes the distribution x?1, then the composition
F(sinh x) exists and given by F(sinh x) = coshec x. Some further similar
results are also deduced. |
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