A Paley-Wiener theorem for locally compact abelian groups
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- by Gunar E. Liepins PDF
- Trans. Amer. Math. Soc. 222 (1976), 193-210 Request permission
Abstract:
Extending the Paley-Wiener theorem to locally compact Abelian groups involves both finding a suitable Laplace transform and a suitable analogue for analytic functions. The Laplace transform is defined in terms of complex characters, and differentiability is defined with use of one-parameter subgroups. The resulting theorem is much as conjectured by Mackey [7],($^{1}$) the major differences being that the theorem is very much an ${L^2}$ theorem and that the problem exhibits a surprising finite dimensional nature.References
-
W. F. Donoghue, Jr., Distributions and Fourier transforms, Academic Press, New York, 1969.
- László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2
- Irving Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. MR 0065561
- Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
- George W. Mackey, The Laplace transform for locally compact Abelian groups, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 156–162. MR 24446, DOI 10.1073/pnas.34.4.156
- Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
- Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019 L. Pontrjagin, Topological groups, Princeton Math. Ser., vol. 2, Princeton Univ. Press, Princeton, N. J., 1939. MR 1, 44. K. A. Ross, See E. Hewitt [4].
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
- Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
- R. Saeks, Resolution space operators and systems, Lecture Notes in Economics and Mathematical Systems, Vol. 82, Springer-Verlag, Berlin-New York, 1973. MR 0465307, DOI 10.1007/978-3-642-80735-0
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 222 (1976), 193-210
- MSC: Primary 43A32
- DOI: https://doi.org/10.1090/S0002-9947-1976-0430679-0
- MathSciNet review: 0430679