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Order and Schwartz distributions†

Published online by Cambridge University Press:  18 May 2009

W. A. Feldman  and
J. F. Porter
W. A. Feldman
Affiliation:
Department of Mathematics, University of Arkansas Fayetteville, Arkansas 72701, U.S.A.
J. F. Porter
Affiliation:
Department of Mathematics, University of Arkansas Fayetteville, Arkansas 72701, U.S.A.
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The space of Schwartz distributions on the unit circle Г in the plane is topologically a considerable generalization of the space of regular, finite Borel measures on Г. However, the order structure of is usually taken to be the same as that of : there are no “positive” distributions which are not measures. This perhaps warrants consideration, since the order structure of generates its topology. In this paper we construct a system of order structures for which is a more natural complement in the intermediate stages to the topology of and which provides an interpretation of with its Schwartz topology as a quotient of a generalized base norm space V′. Where denotes the space of continuous functions on Г with its supremum norm topology, V′ is the dual of . The space contains the infinitely differentiable functions on Г with their usual topology, and (via the pointwise ordering on ) in its product ordering is realized as a generalized order unit space. Some consequences for harmonic functions are discussed.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 1 , January 1977 , pp. 25 - 33
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

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