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On the solution of certain integral equations by generalized functions

Published online by Cambridge University Press:  24 October 2008

J. B. Miller
J. B. Miller
Affiliation:
Australian National UniversityCanberra
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Extract

This note is concerned with a method by which generalized solutions can be shown to exist for certain types of integral equations

where f(x) ∈ L2(0, ∞). The method is briefly this. An extended meaning is given to such equations by using generalized functions of a particular type. Then, if (1) denotes a transformation which has no everywhere-defined inverse in the usual sense, it may be possible to define in the extended sense an inverse transformation

so that if f is given, a generalized function g = can be determined as a solution of (1).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 4 , October 1961 , pp. 767 - 777
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Miller, J. B., Hilbert spaces of generalized functions extending L 2, (I). J. Australian Math. Soc. 1 (1960), 281–98.CrossRefGoogle Scholar
(2)Titchmarsh, E. C., Introduction to the theory of Fourier integrals, 2nd ed. (Oxford, 1948).Google Scholar
(3)Bateman Manuscript Project. Tables of integral transforms (McGraw-Hill, 1954).Google Scholar