Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T07:10:13.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert spaces of tempered distributions, Hermite expansions and sequence spaces

Published online by Cambridge University Press:  20 January 2009

Rainer H. Picard
Rainer H. Picard
Affiliation:
Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukee, Wisconsin 53211, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Although it is well-known that tempered distributions on ℝn can be expanded into series of Herrnite functions, it does not seem to be known, however, that expansions of this type are accessible through the elementary concept of orthonorma! expansions in Hilbert space. This approach is developed here complementing previous work on a Hilbert space approach to distributions. The basis of the development is the observation that the Hermite functions are a complete orthogonal set in each space of a certain scale of Sobolev type Hilbert spaces associated with the family of differential operators defined by

Here Ф denotes a smooth function with compact support. The setting is first developed in the one-dimensional case. By use of the usual multi-index notation this can be extended to the higher-dimensional case. As applications various imbedding results are derived. The paper concludes with a characterization of tempered distributions by convergent Hermite expansions.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 2 , June 1991 , pp. 271 - 293
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Akhiezer, N. I. and Glasman, I. M., Theory of Linear Operators in Hilbert Space, Vol. 1 (Pitman Publ. Ltd., London, 1981).Google Scholar
2.Akhiezer, N. I. and Glasman, I. M., Theory of Linear Operators in Hilbert Space, Vol. 2 (Pitman Publ. Ltd., London, 1981).Google Scholar
3.Antosik, P., Mikusinski, J. and Sikorski, R., Theory of Distributions, The Sequential Approach (Elsevier Scientific Publ. Co., New York, 1973).Google Scholar
4.Bachman, G. and Narici, L., Functional Analysis (Academic Press, New York, 1966).Google Scholar
5.Bonic, R. A., Linear Functional Analysis (Gordon and Breach, New York, 1969).Google Scholar
6.Picard, R., A Hilbert Space Approach to Some Classical Integral Transforms (Pitman Research Notes in Mathematics, Longman, New York, 1989).Google Scholar
7.Picard, R., A Hilbert space approach to distributions. Proc. Roy. Soc. Edinburgh 115A (1990), 275–288.CrossRefGoogle Scholar
8.Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals (Clarendon Press, Oxford, 1948).Google Scholar