Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T16:52:14.148Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results in spectral synthesis

Published online by Cambridge University Press:  24 October 2008

Robert J. Elliott
Robert J. Elliott
Affiliation:
King's College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 2 , April 1965 , pp. 395 - 424
Copyright
Copyright © Cambridge Philosophical Society 1965

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bourbaki, N.Intégration (Hermann; Paris, 1952 and 1956).Google Scholar
(2)Braconnier, J.L'analyse Harmoniques dans les groupes abéliens. Enseignement Math. 2 (1956), 12–41 and 257-273.Google Scholar
(3)Dieudonné, J.Foundations of modern analysis (Academic Press; London, 1960).Google Scholar
(4)Ehrenpreis, L.Solution of some problems of division. Amer. J. Math. 76 (1954), 883–903.CrossRefGoogle Scholar
(5)Ehrenpreis, L.Mean periodic functions. Amer. J. Math. 77 (1955), 293–328.CrossRefGoogle Scholar
(6)Ehrenpreis, L.Appendix to ‘Mean periodic functions’. Amer. J. Math. 77 (1955), 731–733.CrossRefGoogle Scholar
(7)Ehrenpreis, L.Analytic functions and Fourier transforms of distributions. Amer. J. Math. 63 (1956), 129–159.Google Scholar
(8)Hewitt, E. and Ross, K. A.Abstract harmonic analysis, i (Springer; Berlin, 1963).Google Scholar
(9)Kahane, J. P.Sur quelques problèmes d'unicité. Ann. Inst. Fourier (Grenoble), 5 (1954), 39–130.CrossRefGoogle Scholar
(10)Lefranc, M.L'analyse harmonique dans Zn. C.R. Acad. Sci. Paris, 246 (1958), 1951–1953.Google Scholar
(11)Loomis, L.Abstract harmonic analysis (Van Nostrand; Princeton, 1953).Google Scholar
(12)Mackey, G.Laplace transform for locally compact Abelian groups. Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 156–162.CrossRefGoogle ScholarPubMed
(13)Malgrange, B.Existence et approximation des solutions des équations. Ann. Inst. Fourier (Grenoble), 6 (1956), 271–355.CrossRefGoogle Scholar
(14)Malgrange, B. In Seminaire Lelong (Paris, 1959).Google Scholar
(15)Schwartz, L.Théorie générale des fonctions moyennes périodiques. Ann. of Math. 48, (1947), 857–929.CrossRefGoogle Scholar
(16)Schwartz, L.Théorie des distributions, vols. i and ii (Hermann; Paris, 1957).Google Scholar
(17)Weil, A.L'integration dans les groupes topologiques (Hermann; Paris, 1953).Google Scholar