= F. and M. Riesz theorem
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If a finite complex <Borel measure> $\mu$ on the circle has <Fourier coefficients> $\widehat\mu(n)=0$ for all $n<0$, then it is absolutely continuous with respect to arc length. The reflected Fourier convention gives the equivalent theorem with positive coefficients vanishing. The <Poisson integral on the unit disk> is analytic because of the one-sided Fourier spectrum. Subharmonic comparison for its square root and an $L^2$ maximal estimate produce an integrable radial maximum. This maximum dominates the weak-star boundary measure, forcing <absolute continuity of measures>. The same proof on the real line says that a finite measure with <Fourier transform of a finite measure> supported in $[0,\infty)$ has a Lebesgue density.
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