Rational approximation on the union of sets

Abstract:

A counterexample is given to a conjecture of Val’skiǔ that if K is a compact plane set with interior U and the continuous function f on K satisfies $f|\bar U \in R(\bar U)$ and $f|bK \in R(bK)$ then $f \in R(K)$. The conjecture is shown to be true when U is a disc.