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**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)].

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)].

Reviewer: Marcel G. de Bruin (Haarlem)

### 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 |