Annihilating measures of the algebra R(X) (Q801259)

We introduce a special class of ''analytic measures'' in the totality of measures orthogonal to the algebra of rational functions R(X) on a compact set \(X\subset {\mathbb{C}}.\) They are defined as weak* limits of the analytic differentials ''sitting'' on the boundaries of ''nice'' sets converging to X. We show that if \(R(X)\neq C(X),\) then there always exist non-trivial analytic measures. Moreover, if functions harmonic in the neighborhood of X are dense in \({\mathcal C}(\partial X)\), then measures are weak* dense in the space of all annihilating measures for R(X) concentrated on the boundary of X.