## On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions.(English)Zbl 1347.41014

In this paper the authors prove results that were partially announced in their paper [Russ. Math. Surv. 69, No. 1, 159–161 (2014; Zbl 1290.31002); translation from Usp. Mat. Nauk 69, No. 1, 169–170 (2014)].
The main theorems are formulated as
Theorem 1. Let $$E=\cup_{j=1}^m$$, $$E_j\subset{\mathcal E}_m$$, $$f\in{\mathcal A}(E),\mu\in M(E),\mu(E_j)>0\;(1\leq j\leq m)$$. Suppose thet the $$\mu$$-minimizing element in $${\mathcal K}_{E,f}$$ consists of a finite number of continua. Then $$F$$ is symmetric in the external field $${\mathcal V}^{-\mu}$$ and $$\mathbb C\setminus F=\cup_{j=1}^m\,D_j$$, where $$E_j\subset D_j$$.
Moreover, the domains $$D_j$$ and $$D_k\;(1\leq j,k\leq m)$$ either do not intersect one another or coincide.
Theorem 2. Let $$E\subset [a,b]\subset\mathbb C$$ be a closed interval, $$\mu\in{\tilde M}(E)$$. There exists a function $$f\in{\mathcal A}(E)$$ such that the $$\mu$$-minimization problem $\mathrm{cap}_{\mu}\,F=\inf_{K\in {\mathcal K}_{E,f}}\,\mathrm{cap}_{\mu}\,K,$ in the family of compacta $${\mathcal K}_{E,f}$$ is not solvable.
$${\mathcal E}_m$$ is the class of compacta $$E\subset\overline{\mathbb C}$$ of the form $$E=\cup_{j=1}^m\,E_j$$, where $$E_1,\dots,E_m$$ are pairwise disjoint continua in $$\overline{\mathbb C}$$ (some may consist of a single point),
$${\mathcal A}(E)$$ is the class of functions defined on $$E=\cup_{j=1}^m\,E_j$$, such that each restriction $$f_j=f\mid_{E_j}\,(1\leq j\leq m)$$ is holomorphic on $$E_j$$ and admits a continuation along every path in $$\mathbb C$$, not passing through a finite pointset $$A_{f_j}$$, where $$A_{f_j}$$ contains at least one branch point of $$f_j$$…
A very nicely written paper, giving ample background of historical developments and showing in a clear and concise way all the intricate steps in the proofs.

### MSC:

 41A20 Approximation by rational functions 41A21 Padé approximation

Zbl 1290.31002
Full Text:

### References:

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