## Padé-Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $$S$$-property of stationary compact sets.(English. Russian original)Zbl 1243.30076

Russ. Math. Surv. 66, No. 6, 1015-1048 (2011); translation from Usp. Mat. Nauk 66, No. 6, 3-36 (2011).
This elegantly written paper gives an excellent introduction to diagonal Padé-Chebyshev rational approximation for multivalued analytic functions that are real-valued on the unit-interval $$[-1,1]$$, describing the state of art and also presenting new results, with a bibliography containing 79 titles.
For a natural number $$n$$ define the class $${\mathcal R}_n$$ as the collection of all rational functions of the form $$r=p/q$$, where $$\deg p\leq n$$, $$\deg q\leq n$$, $$q\not\equiv 0$$ (an element of this set thus has $$2n+1$$ free parameters). Given a function $$f$$ analytic and real-valued on $$E=[-1,1]$$ that admits a holomorphic continuation (analytic and single-valued) from $$E$$ into some domain $$U\supset E$$, the quantities that are of interest are given by $\varepsilon_n(f)=\inf_{r\in{\mathcal R}_n} ||f-r||_E=\inf_{r\in{\mathcal R}_n} \sup_{x\in E} |(f-r)(x)|.$ This is called the best uniform approximation of $$f$$ on $$E$$ in the class $${\mathcal R}_n$$.
The rate of convergence of the quantity $$\varepsilon_n(f)$$ is studied and characterized in terms of stationary compact sets for the mixed equilibrium problem of Green-logarithmic potentials.
It is not feasible to give all the definitions needed to formulate the known and new results in a short review, but Theorem 6 in the paper can be seen as a generalization and extension of H. Stahl’s ‘second’ theorem [Complex Variables, Theory Appl. 4, 311–324 (1985; Zbl 0542.30027); ibid. 4, 325–338 (1985; Zbl 0542.30028)].

### MSC:

 30E10 Approximation in the complex plane 41A21 Padé approximation 30F99 Riemann surfaces 40A15 Convergence and divergence of continued fractions 41A50 Best approximation, Chebyshev systems

### Citations:

Zbl 0542.30027; Zbl 0542.30028
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