Green equilibrium measures and representations of an external field (Q5957493)

In the papers by V. S. Buyarov and E. A. Rakhmanov there was shown that a continuous function \(Q\) on \(\mathbb{R}\) admits the representation NEWLINE\[NEWLINEQ(x)=\int_0^\infty g_{S_\tau}(x) d\tau NEWLINE\]NEWLINE where \(\{S_\tau\}\) is a certain increasing sequence of compact intervals and \(g_{S_\tau}\) stands for the Green function for \(\mathbb{C}\backslash S_\tau\) with pole at \(\infty\). This representation proved to be of importance in the study of orthogonal polynomials corresponding to the weight \(e^{-Q(x)}\). The authors prove an analogue of this representation which could be useful in a sense for problems involving rational functions. They show that for a certain wide class of functions \(Q(x)\) on a set \(E\) in a domain \(G\subseteq \mathbb{C}\) there exists a suitable increasing family of compact sets \(S_\tau\subseteq E\), \(\tau >0,\) such that for \(t >0\) and \(z\in S_t\) one has the representation NEWLINE\[NEWLINEQ(z)=\int_0^\infty \left\{\frac{1}{\text{cap}_G S_\tau}- V^{\omega_\tau^G}(z)\right\} d\tauNEWLINE\]NEWLINE where \(\text{cap}_G S_\tau\) denotes the Green capacity for the set \(S_\tau\) and \(V^{\omega_\tau^G}(z)\) is the Green potential for the Green equilibrium measure \(\omega_\tau^G\) for \(S_\tau\).