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p-Helson sets, 1<p<2

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Abstract

Ap-Helson set is defined to be a closed subsetE of the circle groupT with the property that every continuous function onE can be extended to the full circle in such a way that this extension has its sequence of Fourier coefficients inl p. For 1<p<2, the union of two such sets is again ap-Helson set. It is shown that thep-Helson sets (p>1) differ from the Helson sets and also that the notion really depends on the indexp. An analogue of H. Helson’s result is given: ap-Helson set supports no nonzero measure with Fourier-Stieltjes transform inl q, 1/p+1/q=1.

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Authors and Affiliations

  1. The University of North Dakota, North Dakota, USA

    Michael B. Gregory

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  1. Michael B. Gregory
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Gregory, M.B. p-Helson sets, 1<p<2. Israel J. Math. 12, 356–368 (1972). https://doi.org/10.1007/BF02764627

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  • DOI: https://doi.org/10.1007/BF02764627

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