### A characterization of algebraic measures

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The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.

For a locally compact, abelian group $G$, we study the space ${S}_{0}\left(G\right)$ of functions on $G$ belonging locally to the Fourier algebra and with ${l}^{1}$-behavior at infinity. We give an abstract characterization of the family of spaces $\left\{{S}_{0}\right(G):G$ abelian$\}$ by its hereditary properties.

Suppose that $G$ is a locally compact abelian group with a Haar measure $\mu $. The $\delta $-ball ${B}_{\delta}$ of a continuous translation invariant pseudo-metric is called $d$-dimensional if $\mu \left({B}_{2{\delta}^{\prime}}\right)\le {2}^{d}\mu \left({B}_{{\delta}^{\prime}}\right)$ for all ${\delta}^{\prime}\subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu \left(nA\right)\le {n}^{d}\mu \left(A\right)$ for all $n\ge dlogd$, then $A$ is contained in an $O\left(d{log}^{3}d\right)$-dimensional ball, $B$, of positive radius in some continuous translation invariant pseudo-metric and $\mu \left(B\right)\le exp\left(O\right(dlogd\left)\right)\mu \left(A\right)$.

Let $\mu $ be a Fourier-Stieltjes transform, defined on the discrete real line and such that the corresponding measure on the dual group vanishes on the set of characters, continuous on $\mathbf{R}$. Then for every $\u03f5\>0$, $\{x\in \mathbf{R}|\phantom{\rule{0.166667em}{0ex}}\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\left(\mu \right(x\left)\right)\>\u03f5\}$ has a vanishing interior Lebesgue measure. If $\u03f5=0$ the statement is not generally true. The result is applied to prove a theorem of Rosenthal.

We give an elementary proof for the case of the circle group of the theorem of O. Hatori and E. Sato, which states that every measure on a compact abelian group G can be decomposed into a sum of two measures with a natural spectrum and a discrete measure.