Reconstruction of holomorphic functions of several variables from their values on a uniqueness set
A compact \(K\subset \bar D\), where \(\bar D\) is the closure of a domain \(D\subset {\mathbb{C}}^ n\), is said to be a uniqueness set for the class A(D) of the functions continuous in \(\bar D\) and holomorphic in D if \(f\in A(D)\) and \(f=0\) on K imply \(f=0\) on \(\bar D.\) A compact K has the property of conditional stability if for each sequence \(\{f_ n\}\), \(f_ n\in A(D)\), \(| f_ n| \leq M\) in D and \(f_ n\to 0\) uniformly on K, then \(f_ n(z)\to 0\) for each \(z\in D\). The authors give a negative answer, by means of two counter-examples, to the following open question raised by \textit{L. A. Ajzenberg} and \textit{N. N. Tarkhanov} [Sov. Math., Dokl. 37, No.1, 235-238 (1988); translation from Dokl. Akad. Nauk SSSR 298, No.6, 1292-1296 (1988)]: is each uniqueness compact set a compact set satisfying the property of conditional stability? Or, in other words, given a uniqueness compact set, is it possible to reconstruct a function of A(D) by means of a formula of Carleman's type?
reconstruction of a holomorphic function
scientific article; zbMATH DE number 4208482