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On the Stieltjes transform of generalized functions

Published online by Cambridge University Press:  24 October 2008

J. N. Pandey
J. N. Pandey
Affiliation:
Carleton University, Ottawa, Canada
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Extract

If f(t) belongs to L(0, R) for every positive R and is such that the integral

converges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis and

for any positive ξ at which f(ξ+) and f(ξ−) both exist.

We define an operator Lk, t[F(x)]by

Under the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),

Let Ln, x denote the differentiation operator

Suppose that

converges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 1 , January 1972 , pp. 85 - 96
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

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