Two footnotes to the F. & M. Riesz theorem

TL;DR

This paper introduces a new proof of the F. & M. Riesz theorem on analytic measures on the unit circle, utilizing an elementary inequality and extending the approach to the infinite-dimensional torus, clarifying related criteria.

Contribution

It provides a novel proof based on an elementary inequality and extends the results to infinite-dimensional settings, linking Hilbert's criterion with the theorem.

Findings

New proof of F. & M. Riesz theorem using elementary inequality

Extension of proof to infinite-dimensional torus

Clarification of Hilbert's criterion relationship

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 $0 \leq r \leq \varrho < 1$, then \[\|f_r-f_\varrho\|_1 \leq 2 \sqrt{\|f_\varrho\|_1^2-\|f_r\|_1^2},\] where $f_r(e^{i\theta})=f(r e^{i\theta})$ and where $\|\cdot\|_1$ 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.