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On Operators and Distributions

Published online by Cambridge University Press:  20 November 2018

Raimond A. Struble
Raimond A. Struble*
Affiliation:
North Carolina State University
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Mikusinski [1] has extended the operational calculus by methods which are essentially algebraic. He considers the family C of continuous complex valued functions on the half-line [0,∞). Under addition and convolution C becomes a commutative ring. Titchmarsh's theorem [2] shows that the ring has no divisors of zero and, hence, that it may be imbedded in its quotient field Q whose elements are then called operators. Included in the field are the integral, differential and translational operators of analysis as well as certain generalized functions, such as the Dirac delta function. An alternate approach [3] yields a rather interesting result which we shall now describe briefly.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 1 , April 1968 , pp. 61 - 64
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Mikusinski, J., Operational Calculus, Pergamon Press, London and New York, 1959.Google Scholar
2. Mikusinski, J., ibid, 15-23.Google Scholar
3. Koh, Kwangil and Struble, Raimond A., An Approach to the Operational Calculus,Unpublished manuscript, 1965.Google Scholar
4. Johnson, R.E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895.10.1090/S0002-9939-1951-0045695-9Google Scholar
5. Findlay, G.D. and Lambek, J., A generalized ring of quotients II, Canad. Math. Bull., Vol. 1, No. 3, Sept. 1958, 155-167.10.4153/CMB-1958-016-6Google Scholar
6. Zemanian, A.H., Distribution Theory and Transform Analysis, McGraw-Hill, New York, 1965.Google Scholar
7. Schwartz, L., Théorie des Distributions, Tome I. Paris, Hermann, 1957.Google Scholar
8. Norris, D.O., A Topology for Mikusinski Operators, Studia Math. 24(1964), 245-255.10.4064/sm-24-3-245-255Google Scholar