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Entire Mean Periodic Functions

Published online by Cambridge University Press:  20 November 2018

P. G. Laird
P. G. Laird*
Affiliation:
The University of Wollongong, Wollongong, N.S.W. 2500, A ustralia
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Let H denote the set of all entire functions of a single complex variable equipped with the topology of convergence uniform on all compact subsets of C, the set of complex numbers. Then an entire function f is mean periodic if the subspace spanned by f and its complex translates is not dense in H. It was shown by Schwartz [13, p. 922] in 1947, to whom this definition is due, that any such function is the limit in H of a certain sequence of exponential polynomials.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 4 , 01 August 1975 , pp. 805 - 818
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Delsarte, J., Théorie des fonctions moyenne périodiques de deux variables, Ann. of Math. 72 (1960), 121–178.Google Scholar
2. Delsarte, J., Lectures on topics in mean-periodic functions and the two radius problem (Tata institute, Bombay, 1961).Google Scholar
3. Ehrenpreis, L., Mean periodic functions, I, Amer. J. Math. 77 (1955), 293–328.Google Scholar
4. Erdélyi, A., Operational calculus and generalized functions (Holt, Rinehart and Winston, 1962).Google Scholar
5. Kahane, J. P., Sur quelques problèmes d'unicité et de prolongement, relatifs aux fonctions approchable par des sommes d'exponentielles, Ann. Inst. Fourier (Grenoble) 5 (1953-54), 39–130.Google Scholar
6. Kahane, J. P., Lectures on mean periodic functions (Tata Institute, Bombay, 1959).Google Scholar
7. Laird, P. G., Some properties of mean periodic functions, J. Austral. Math. Soc. 14 (1972), 424–432.Google Scholar
8. Laird, P. G., Mean periodic functions and functional differential equations, Ph.D. Thesis, University of Calgary, 1973.Google Scholar
9. Laird, P. G., Ideals in a ring of exponential polynomials, J. Austral. Math. Soc. 17 (1974), 257–262.Google Scholar
10. Laird, P. G., Functional differential equations and continuous mean periodic functions, J. Math. Anal. Appl. 47 (1974), 406–423.Google Scholar
11. Malgrange, B., Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 271–355.Google Scholar
12. Rubel, L. A., personal communication.Google Scholar
13. Schwartz, L., Théorie générale des fonctions moyenne périodiques, Ann. of Math. 48 (1947), 857–929.Google Scholar
14. Titchmarsh, E. C., Introduction to the theory of Fourier integrals (Oxford Univ. Press, Oxford, 1948).Google Scholar