Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Unique balayage in Fourier transforms on compact abelian groups

Author: George S. Shapiro
Journal: Proc. Amer. Math. Soc. 70 (1978), 146-150
MSC: Primary 43A05
MathSciNet review: 0477600
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let K be a compact subset of the compact abelian group G and let $ \Lambda $ be a subset of the dual group $ \Gamma $. Unique balayage is said to be possible for $ (K,\Lambda )$ if, for every $ \mu $ in $ M(G)$, there is a unique $ \nu $ in $ M(K)$ whose Fourier transform, $ \hat \nu $, agrees on $ \Lambda $ with $ \hat \mu $.

We prove that in order that there be any K with unique balayage possible for $ (K,\Lambda ),\Lambda $ must belong to the coset ring of $ \Gamma $. The converse of this statement is false. Some examples are given for the case where G is the circle group.

References [Enhancements On Off] (What's this?)

  • [1] John J. Benedetto, Spectral synthesis, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, No. 66. MR 0622040
  • [2] A. Beurling, On balayage of measures in Fourier transforms, Seminar notes, Institute for Advanced Study, Princeton, N. J., 1959-1960 (unpublished).
  • [3] N. Dunford and J. T. Schwartz, Linear operators.I, Interscience, New York, 1958.
  • [4] Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1972. North-Holland Mathematical Library, Vol. 2. MR 0485769
  • [5] Haskell P. Rosenthal, Projections onto translation-invariant subspaces of 𝐿^{𝑝}(𝐺),𝐺 non-compact, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 265–275. MR 0196421
  • [6] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 43A05

Retrieve articles in all journals with MSC: 43A05

Additional Information

Keywords: Balayage in Fourier transforms, compact abelian groups, idempotent measures, coset ring
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society