An analogue of Montel's theorem for rational functions of best \(L_p\)-approximation
For a domain \(B\subset \mathbb{C}\) denote by \(\mathcal A(B)\) the set of complex-valued analytic functions on \(B\), and for \(M\subset B\) let \(\mathcal A_M(B)\) be the set of functions on \(M\) which admit analytic extensions to \(B\). Denote by \(\nu (g,M)\) the number of zeros in the set \(M\) of a complex function \(g\) defined on \(M\). Suppose that \(S\) is a regular continuum [see Chapter V in \textit{G. M. Goluzin}, Geometric theory of functions of a complex variable (1969; Zbl 0183.07502)] and \(B\supset S\) is a domain in the complex plane \(\mathbb{C}\). Let \(F = \{f_n : n=1, 2, \dots \} \subset \mathcal A(B)\) be a sequence of rational functions with a total number of poles in \(\bar {\mathbb{C}}\) of every \(f_n\) not exceeding \(n\), and with \(\nu (f_n,K) = o(n)\) for \( n \to \infty. \) If \(f\in \mathcal A(S)\) is such that \(\limsup _n\|f_n-f\|^{1/n}<1\) then the sequence \(F\) converges locally uniformly inside \(B\) and \(f\in \mathcal A_S(B)\). This result is applied to sequences of rational functions of best \(L_p\)-approximation with an unbounded number of finite poles. In the case of polynomials a similar problem was considered by \textit{H.-P. Blatt, E. B. Saff} and \textit{M. Simkani} [J. Lond. Math. Soc., II Ser. 38, No. 2, 307-316 (1988; Zbl 0684.41006)] and by \textit{R. K. Kovacheva}, Math. Balk., New Ser. 3/4, 244-256 (1989; Zbl 0716.30031)] for meromorphic functions with not more than \(m\) poles and rational functions with the degree of the denominator also \(m\).
scientific article; zbMATH DE number 1750340
scientific article