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A Hilbert space approach to distributions

Published online by Cambridge University Press:  14 November 2011

Rainer H. Picard
Affiliation:
Department of Mathematical Sciences, University of Wisconsin – Milwaukee, Milwaukee, WI 53211, U.S.A.

Synopsis

A compact chain of Sobolev type Hilbert spaces , n integer, is introduced that is invariant with

respect to the Fourier transform ℱ. The spaces are related to powers of the adjoint of the so-called tempered derivative introduced in the sequential approach to distributions. It turns out that the intersection of all these Hilbert spaces coincides with the space of rapidly decaying C-functions and their union leads to the space of tempered distributions. Moreover, the naturally induced convergence concepts coincide with the usual ones. The approach provides not only a new and arguably more elementary approach to distributions it also provides a deeper insight into the action of the Fourier transform which is a unitary mapping in each space of the chain. Finally the Schwartz distributions are incorporated in the approach as locally tempered distributions.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1990

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