On meromorphic approximation (Q5934408)

The symbol \({\mathcal M}_{n,m}\) denotes here a class of meromorphic functions on a bounded domain \(G\), \(0\in G\), with boundary \(\Gamma\) consisting of \(N\) disjoint closed analytic Jordan curves. More precisely, the functions of the class \({\mathcal M}_{n,m}\) are represented in the form \(h=p/qz^m\), where \(p\in E_\infty (G)\) (Smirnov class on \(G)\), \(q\) is a polynomial of degree at most \(n\), \(q \equiv 0\). The main purpose of this paper is to obtain necessary and sufficient conditions for a function \(h\) of the class \({\mathcal M}_{m,n}\) to be an element of best approximation to a continuous function \(f\) on \(\Gamma\), in the space \(L_\infty (\Gamma)\) (Theorems 1 and 2). Some applications of this results are shown. Let \(h_{n,m}\) be an element of the best approximation to \(f\). Theorem 1 takes into account the representation \(h_{n,m}= P/Q\xi^m\), where \(\deg Q=n-d\) and \(d\) is the defect of \(h_{n,m}\), to assert that there exists a function \(\varphi\in E_1(G)\), \(\varphi\equiv 0\), with at least \(d\) zeros in \(G\), such that \[ Q^2(\xi)\xi^m \varphi(\xi)(f-h_{n,m}) (\xi)d\xi= \Delta_{n,m} \bigl |Q^2(\xi) \xi^m\varphi (\xi)\bigr||d\xi |\tag{1} \] almost everywhere on \(\Gamma\). In the paper equation (1) is applied for proving a generalization of the Adamyan-Arov-Kreĭn theorem. The last section is devoted to study the connection of this theory with the orthogonal polynomials. For the case when \(G\) is the unit disk it is also proved from (1) that the polynomial \(Q_n\) constructed from the nonzero poles of \(h_{n,m}\) is the \(n\)th orthogonal polynomial with respect to a complex-valued measure \(d\mu_n(\xi)\), \(\xi\in\gamma\), where \(f\) is now assumed to be holomorphic on \(\overline\mathbb{C} \setminus E\), \(E\) is a compact set in \(G\), and \(\gamma\subset G\) surrounds \(E\). The case when \(G\) is an \(N\)-connected domain is also considered.