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The Gonchar-Stahl \(\rho^2\)-theorem and associated directions in the theory of rational approximations of analytic functions. (English. Russian original) Zbl 1361.30061

Sb. Math. 207, No. 9, 1236-1266 (2016); translation from Mat. Sb. 207, No. 9, 57-90 (2016).
Let \(f\) be an analytic function on a continuum \(E\) (with \(\hat{\mathbb C}\setminus{E}\) connected, \(\hat{\mathbb C}:=\mathbb C\cup\{\infty\}\)) which admits analytic continuation along any path in a domain \(\mathcal D\) with \(E\subset\mathcal D\) and \(\mathrm{cap}(\hat{\mathbb C}\setminus\mathcal D)=0\). Further, let \(\rho_n(f,E)\) be the supremum norm of the best rational approximation of \(f\), that is, \[ \rho_n(f,E):=\min\sup_{z\in{E}}|f(z)-r(z)|, \] where the minimum is taken over all rational functions \(r=P_n/Q_n\) and \(P_n,Q_n\) polynomials of degree at most \(n\). Finally, define \[ \rho(f,E):=\inf\mathrm e^{-1/\mathrm{cap}(E,F)}, \] where \(\mathrm{cap}(E,F)\) is the capacity of the condenser \((E,F)\) and the infimum is taken over all sets \(F=\partial\Omega\) with \(E\subset\Omega\) and \(f\) holomorphic on \(\Omega\). One of the central results in the theory of rational approximation of analytic functions is the so-called Gonchar-Stahl \(\rho^2\)-theorem, which says that \[ \lim_{n\to\infty}\rho_n(f,E)^{1/n}=\rho(f,E)^2. \] In this survey article, the author gives a thorough insight into the theory of rational approximation of analytic functions, both from a historical point of view starting from a well-known theorem of Walsh in 1930’s and also by describing the various methods (which are interesting for themselves) used in the proof. Furthermore, the connection to Padé approximation of functions with branch points and some generalizations and conjectures are discussed.

MSC:

30E10 Approximation in the complex plane
30C85 Capacity and harmonic measure in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
41A20 Approximation by rational functions
41A25 Rate of convergence, degree of approximation
41A21 Padé approximation
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References:

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