## 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:

  Aptekarev, A. I., Asymptotics of Hermite-Padé approximants for a pair of functions with branch points (Russian), Dokl. Akad. Nauk, 422, 4, 443-445, (2008), translation in: Dokl. Math. 78 (2) (2008) 717-719  Alexander I. Aptekarev, Maxim L. Yattselev, Padé approximants for functions with branch points—strong asymptotics of Nuttall-Stahl polynomials, arXiv: http://arxiv.org/abs/1109.0332 (2011), 45 pp.  Baratchart, L.; Stahl, H.; Yattselev, M., Weighted extremal domains and best rational approximation, Adv. Math., 229, 1, 357-407, (2012) · Zbl 1232.41014  Buslaev, V. I., On the Convergence of Continued T-Fractions, Analytic and geometric issues of complex analysis, Collected papers. Dedicated to the 70th anniversary of academician Anatolii Georgievich Vitushkin, Tr. Mat. Inst. Steklova, vol. 235, 36-51, (2001), Nauka Moscow, transl.: Proc. Steklov Inst. Math. 235 (2001) 29-43  Buslaev, V. I., Convergence of multipoint Padé approximants of piecewise analytic functions, Mat. Sb., 204, 2, 39-72, (2013), transl.: Sb. Math. 204 (2) (2013) 190-222 · Zbl 1276.41011  Buslaev, V. I., The convergence of $$m$$-point Padé approximants to the set of multi-valued analytic functions, Mat. Sb., 206, 2, 5-30, (2015)  Buslaev, V. I.; Suetin, S. P., Existence of compact sets with minimum capacity in problems of rational approximation of multi-valued analytic functions, Uspekhi Mat. Nauk, 69, 1(415), 169-170, (2014), transl: Russian Math. Surveys 69 (1) (2014) 159-161 · Zbl 1290.31002  Buslaev, V. I.; Martínez-Finkelshtein, A.; Suetin, S. P., Method of interior variations and existence of $$S$$-compact sets, (Analytic and Geometric Issues of Complex Analysis, Collected Papers, Tr. Mat. Inst. Steklova, vol. 279, (2012), MAIK Nauka/Interperiodica Moscow), 31-58, transl: Proc. Steklov Inst. Math. 279 (2012) 25-51 · Zbl 1298.30028  Deano, A.; Huybrechs, D.; Kuijlaars, A. B.J., Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature, J. Approx. Theory, 162, 12, 2202-2224, (2010) · Zbl 1223.41017  Gonchar, A. A., Rational approximations of analytic functions, Sovrem. Probl. Mat., 1, 83-106, (2003), Steklov Math. Inst., RAS, Moscow; transl: Proc. Steklov Inst. Math. 272 (suppl. 2 ) (2011) S44-S57 · Zbl 1355.30036  Gonchar, A. A.; Rakhmanov, E. A., Equilibrium distributions and degree of rational approximation of analytic functions, Mat. Sb. (N.S.), 134(176), 3(11), 306-352, (1987), transl: Math. USSR-Sb. 62 (2) (1989) 305-348 · Zbl 0645.30026  Gonchar, A. A.; Rakhmanov, E. A.; Suetin, S. P., Padé-Chebyshev approximants of multi-valued analytic functions, variation of equilibrium energy, and the $$S$$-property of stationary compact sets, Uspekhi Mat. Nauk, 66, 6(402), 3-36, (2011), transl: Russian Math. Surveys 66 (6) (2011) 1015-1048 · Zbl 1243.30076  Grötzsch, H., Uber ein variationsproblem der konformen abbildung, Berichte Leipzig, 82, 251-263, (1930) · JFM 56.0298.03  Komlov, A. V.; Suetin, S. P., An asymptotic formula for a two-point analogue of Jacobi polynomials, Uspekhi Mat. Nauk, 68, 4(412), 183-184, (2013), transl: Russian Math. Surveys 68 (4) (2013) 779-781 · Zbl 1284.41006  Kuijlaars, A. B.J.; Silva, G. L.F., $$S$$-curves in polynomial external fields, J. Approx. Theory, 191, 1-37, (2015) · Zbl 1314.31006  Kuz’mina, G. V., Moduli of families of curves and quadratic differentials, Tr. Mat. Inst. Steklova, 139, 3-241, (1980), transl: Proc. Steklov Inst. Math. 139 (1982) 1-231 · Zbl 0482.30015  Kuz’mina, G. V., Gennadii mikhailovich goluzin and geometric function theory, Algebra i Analiz, 18, 3, 3-38, (2006), transl: St. Petersburg Math. J. 18 (3) (2007) 347-372 · Zbl 1137.01022  Landkof, N. S., Osnovy sovremennoi teorii potentsiala [fundamentals of modern potential theory], (Russian, Izdat, (1966), Nauka’ Moscow)  Lavrentieff (Lavrentiev), M., Sur un probleme de maximum dans la representation conforme, C. R., 191, 827-829, (1930) · JFM 56.0299.02  Lavrentieff (Lavrentiev), M., Zur theorie der konformen abbildungen, Russian, Trav. Inst. Phys.-Math. Stekloff, 5, 159-245, (1934)  Martínez-Finkelshtein, A.; Rakhmanov, E., Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials, Comm. Math. Phys., 302, 1, 53-111, (2011) · Zbl 1226.30005  Martínez-Finkelshtein, A.; Rakhmanov, E. A.; Suetin, S. P., Variation of the equilibrium energy and the $$S$$-property of stationary compact sets, Mat. Sb., 202, 12, 113-136, (2011), transl: Sb. Math. 202 (12) (2011) 1831-1852 · Zbl 1244.31001  Martínez-Finkelshtein, A.; Rakhmanov, E. A.; Suetin, S. P., Heine, Hilbert, Padé, Riemann, and Stieltjes: a John nuttall’s work 25 years later, (Recent Advances in Orthogonal Polynomials, Special Functions, and their Applications, Contemp. Math., vol. 578, (2012), Amer. Math. Soc. Providence, RI), 165-193 · Zbl 1318.42033  Nuttall, J., Asymptotics of generalized Jacobi polynomials, Constr. Approx., 2, 1, 59-77, (1986) · Zbl 0585.41014  E. Perevoznikova, E. Rakhmanov, Variations of the equilibrium energy and $$S$$-property of compacta of minimal capacity, Manuscript (1994).  Rakhmanov, E. A., Orthogonal polynomials and $$S$$-curves, recent advances in orthogonal polynomials, special functions, and their applications, (Contemp. Math., vol. 578, (2012), Amer. Math. Soc Providence, RI), 195-239 · Zbl 1318.30056  Rakhmanov, E. A.; Suetin, S. P., The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a nikishin system, Mat. Sb., 204, 9, 115-160, (2013), transl: Sb. Math. 204 (9) (2013) 1347-1390 · Zbl 1288.26010  Saff, E. B.; Totik, V., Logarithmic potentials with external fields, (Appendix B by Thomas Bloom, Grundlehren der Mathematischen Wissenschaften, vol. 316, (1997), Springer-Verlag Berlin) · Zbl 0881.31001  Stahl, H., Extremal domains associated with an analytic function. I, Complex Variables Theory Appl., 4, 311-324, (1985) · Zbl 0542.30027  Stahl, H., Extremal domains associated with an analytic function. II, Complex Variables Theory Appl., 4, 325-338, (1985) · Zbl 0542.30028  Stahl, H., Structure of extremal domains associated with an analytic function, Complex Variables Theory Appl., 4, 339-354, (1985) · Zbl 0542.30029  Stahl, H., Orthogonal polynomials with complex valued weight function. I, Constr. Approx., 2, 225-240, (1986) · Zbl 0592.42016  Stahl, H., Orthogonal polynomials with complex valued weight function. II, Constr. Approx., 2, 241-251, (1986) · Zbl 0606.42021  Stahl, H., Convergence of rational interpolants, numerical analysis (Louvain-la-Neuve, 1995), Bull. Belg. Math. Soc. Simon Stevin, 11-32, (1996), no. suppl  Strebel, K., Quadratic differentials, 5, (Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], (1984), Springer-Verlag Berlin) · Zbl 0547.30001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.