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A convolution-integral representation for a class of linear operators

Published online by Cambridge University Press:  24 October 2008

K. Rowlands
K. Rowlands
Affiliation:
Department of Pure Mathematics, University College of Wales, Aberystwyth
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Extract

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formula

A linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:

(a) for each x in , the complex-conjugate function x* is in ;

(b) both and \ are invariant under the shift operators;

(c) every shift operator commutes with T.

(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 1 , January 1972 , pp. 51 - 60
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

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