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Some sets obeying harmonic synthesis

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Abstract

LetX be a (not necessarily closed) subspace of the dual spaceB * of a separable Banach spaceB. LetX 1 denote the set of all weak* limits of sequences inX. DefineX a , for every ordinal numbera, by the inductive rule:X a = (U b < a X b ) 1 .There is always a countable ordinala such thatX a is the weak* closure ofX; the first sucha is called theorder ofX inB *.

LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa.

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References

  1. J.-P. Kahane,Séries de Fourier absolument convergentes, Ergebnisse der Math. und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin and New York, 1970.

    MATH  Google Scholar 

  2. Y. Katznelson and O. C. McGehee,Some Banach algebras associated with quotients of L 1 (R), Indiana Univ. Math. J. (1971), 419–436.

  3. Th. Körner,A pseudofunction on a Helson set, Astérisque (Soc. Math. France)5 (1973), 3–224.

    MATH  Google Scholar 

  4. O. C. McGehee,A proof of a statement of Banach about the weak * topology, Michigan Math. J.15 (1968), 135–140.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Rudin,Fourier Analysis on Groups, Wiley, New York, 1962.

    MATH  Google Scholar 

  6. D. Sarason,On the order of a simply connected domain, Michigan Math. J.15 (1968), 129–133.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Schneider,Some theorems in Fourier analysis on symmetric sets, Pacific. J. Math.31 (1969), 175–195.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Hebrew University of Jerusalem, Jerusalem, Israel

    Yitzhak Katznelson & O. Carruth McGehee

  2. Louisiana State University, Baton Rouge, Louisiana, U.S.A.

    Yitzhak Katznelson & O. Carruth McGehee

Authors
  1. Yitzhak Katznelson
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  2. O. Carruth McGehee
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Additional information

This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.

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Katznelson, Y., McGehee, O.C. Some sets obeying harmonic synthesis. Israel J. Math. 23, 88–93 (1976). https://doi.org/10.1007/BF02757235

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  • DOI: https://doi.org/10.1007/BF02757235

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