Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-08T04:46:58.690Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Widder, D. V.The Laplace transform (Princeton, 1946).Google Scholar
(2)Widder, D. V.A transform related to the Poisson integral for a half plane. Duke 33, 355–362.Google Scholar
(3)Widder, D. V.The Stieltjes transform. Trans. Amer. Math. Soc. 43, 7–60.Google Scholar
(4)Haimo, D. T.Integral equations associated with Hankel convolutions, Trans. Amer. Math. Soc. 116 (1965), 330–375.Google Scholar
(5)Cholewinski, , Frank, M.A Hankel convolution complex inversion theory. Mem. Amer. Math. Soc. 58 (1965).Google Scholar
(6)Pandey, J. N. & Zemanian, A. H.Complex inversion for the generalized convolution transformation. Pacific 25 (1968), 147–157.Google Scholar
(7)Pandey, J. N.An extension of Haimo's form of Hankel convolutions, Pacific J. Math. 28 (1969), 641–651.Google Scholar
(8)Pandey, J. N. Complex inversion for the generalized Hankel convolution transformation. SIAM J. Appl. Math. (to appear).Google Scholar
(9)Pandey, J. N. The generalized Weierstrass Hankel convolution transform. SIAM J. Appl. Math. (to appear).Google Scholar
(10)Pandey, J. N.On the Stieltjes transform of generalized functions (Carleton Maths Series, 1970).Google Scholar
(11)Zemanian, A. H.Distribution theory and transform analysis (McGraw Hill Book Co., New York, 1965).Google Scholar
(12)Zemanian, A. H.Generalized integral transformations (Interscience Publishers, 1968).Google Scholar
(13)Zemanian, A. H.A generalized convolution transformation, SIAM J. Appl. Math. 15, (03 1967), 324–346.CrossRefGoogle Scholar
(14)Ditzian, Z.Generalized functions and convolution transforms. J. Math. Anal. Appl. 26, (05 1969), 345–356.CrossRefGoogle Scholar
(15)Schwartz, L.Théorié des Distributions, Vol. i and ii (Hermann, Paris 1957, 1959).Google Scholar
(16)Gel'fand, I. M. & Shilov, G. E.Generalized functions, Vol. ii.Google Scholar