# Riesz theorem

Riesz's uniqueness theorem for bounded analytic functions: If $f( z)$ is a bounded regular analytic function in the unit disc $D = \{ {z \in \mathbf C } : {| z | < 1 } \}$ having zero radial boundary values (cf. Radial boundary value) on a subset $E$ of the circle $\Gamma = \{ {z } : {| z | = 1 } \}$ of positive measure, $\mathop{\rm mes} E > 0$, then $f( z) \equiv 0$. The theorem was formulated and proved by the brothers F. Riesz and M. Riesz in 1916 (see ).

This theorem is one of the first boundary value theorems on the uniqueness of analytic functions. Independently of the brothers Riesz, general boundary value theorems on uniqueness were obtained by N.N. Luzin and I.I. Privalov (see , , and Luzin–Privalov theorems).

Riesz's theorem on the Cauchy integral: If $f( z)$ is a Cauchy integral,

$$f( z) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{f( \zeta ) d \zeta }{\zeta - z } ,$$

in the unit disc $D$ and its boundary values $f( \zeta ) = f( e ^ {i \theta } )$ form a function of bounded variation on $\Gamma$, then $f( \zeta )$ is an absolutely-continuous function on $\Gamma$( see ).

This theorem can be generalized to Cauchy integrals along any rectifiable contour $\Gamma$( see ).

How to Cite This Entry:
Riesz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_theorem&oldid=48569
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article