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An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness

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Let E be a compact perfect subset of the real line R such that the restriction of the Fourier transform a bar right arrow (a) over cap vertical bar(E) from L(1)(R) into C(E) is onto. Helson proved that then, for mu is an element of M(E), lim(vertical bar y vertical bar ->infinity vertical bar) (u) over cap (y)vertical bar = 0 is possible only if mu = 0. In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness.

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Academic Press Inc Elsevier Science

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Mathematics

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Journal of Functional Analysis

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10.1016/j.jfa.2009.09.021

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