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 \(0 \leq r \leq \varrho < 1\), 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.