Two footnotes to the F. & M. Riesz theorem

Abstract

We present a new proof of the F. & M. Riesz theorem on analytic measures of the unit circle $\mathbb{T}$ that is based the following elementary inequality: If $f$ is analytic in the unit disc $\mathbb{D}$ and $0\le r\le \varrho <1$, then \[\|f_r-f_\varrho\|_1 \leq 2 \sqrt{\|f_\varrho\|_1^2-\|f_r\|_1^2},\] where $f_r(e^{i\hat{I}¸})=f(r{e}^{i\hat{I}¸})$ and where $\Vert \cdot {\Vert }_1$ denotes the norm of $L^1(\mathbb{T})$. The proof extends to the infinite-dimensional torus ${\mathbb{T}}^{\infty }$, where it clarifies the relationship between Hilbert's criterion for $H^1({\mathbb{T}}^{\infty })$ and the F. & M. Riesz theorem.

This note has been accepted for publication in Annales Fennici Mathematici