Fourier Transforms of Unbounded Measures

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Abstract
1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R: (1.1) More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function (1.2) where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that (1.3) for all ϕ ∈ L1(G) where

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