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Transactions of the American Mathematical Society

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Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies


Author: George S. Shapiro
Journal: Trans. Amer. Math. Soc. 225 (1977), 183-198
MSC: Primary 43A25; Secondary 42A44
DOI: https://doi.org/10.1090/S0002-9947-1977-0425510-4
MathSciNet review: 0425510
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Lambda $ be a discrete subset of an LCA group and E a compact subset of the dual group. Balayage is said to be possible for $ (\Lambda ,E)$ if the Fourier transform of each measure on G is equal on E to the Fourier transform of some measure supported by $ \Lambda $. Following Beurling, we show that this condition is equivalent to the possibility of bounding certain functions with spectra in E by their bounds on $ \Lambda $. We derive consequences of this equivalence, among them a necessary condition on $ \Lambda $ for balayage when E is compact and open (a condition analogous to a density condition Beurling and Landau gave for balayage in Euclidean spaces).

We show that if balayage is possible for $ (\Lambda ,E)$ and if $ \Lambda '$ is close to $ \Lambda $, then balayage is possible for $ (\Lambda ',E)$. Explicit bounds for the needed closeness in R and $ {R^n}$ are given.

Using these perturbation techniques, we give examples of perfect sets $ E \subset R$ with the property that there are ``arbitrarily sparse'' sets $ \Lambda $ with balayage possible for $ (\Lambda ,E)$.


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

  • [1] W. G. Bade and P. C. Curtis, Embedding theorems for commutative Banach algebras, Pacific J. Math. 18 (1966), 391-409. MR 34 # 1878. MR 0202001 (34:1878)
  • [2] A. Beurling, On balayage of measures in Fourier transforms, Notes from a seminar at the Institute for Advanced Study, Princeton, N.J., 1959-60 (unpublished).
  • [3] -, Local harmonic analysis with some applications to differential operators, Annual Science Conference Proceedings, vol. I: 1962-1964, Belfer Graduate School of Science, Academic Press, New York, 1966, pp. 109-125. MR 0427956 (55:986)
  • [4] E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, II, Grundlehren math Wiss., Bände 115, 152, Springer-Verlag, Berlin and New York, 1963, 1970. MR 28 #158; 41 #7378. MR 0262773 (41:7378)
  • [5] J.-P. Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin and New York, 1970. MR 43 #801. MR 0275043 (43:801)
  • [6] J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Actualités Sci. Indust., No. 1301, Hermann, Paris, 1963. MR 28 #3279. MR 0160065 (28:3279)
  • [7] H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37-52. MR 36 #5604. MR 0222554 (36:5604)
  • [8] Y. Meyer, Algebraic numbers and harmonic analysis, North-Holland Math. Library, no. 2, North-Holland, Amsterdam, 1972. MR 0485769 (58:5579)
  • [9] H. Reiter, Classical harmonic analysis and locally compact groups, Oxford Math. Monographs, Clarendon Press, Oxford, London, 1968. MR 46 #5933. MR 0306811 (46:5933)
  • [10] G. S. Shapiro, Some aspects of balayage of Fourier transforms, Dissertation, Harvard Univ., 1973.
  • [11] N. Varopoulos, Sets of multiplicity in locally compact abelian groups, Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 2, 123-158. MR 35 #3379. MR 0212508 (35:3379)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0425510-4
Keywords: Balayage in Fourier transforms, convolution, set of sampling, dual Banach spaces, surjective linear operator, density, uniformly discrete, relatively dense, compact open subgroup, sparse set, Dirichlet set, homogeneous perfect set
Article copyright: © Copyright 1977 American Mathematical Society

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