## On the asymptotics of polynomials orthogonal on a system of curves with respect to a measure with discrete part.(English. Russian original)Zbl 1197.42014

St. Petersbg. Math. J. 21, No. 2, 217-230 (2010); translation from Algebra Anal. 21, No. 2, 71-91 (2009).
Let $$E=\bigcup_{k=1}^p E_k$$ be a union of complex, mutually exterior and rectifiable Jordan curves and arcs, each of which is assumed to be of $$C^{2+}$$ class. Let $$\Omega$$ denote the connected component of $$\mathbb{C}\backslash E$$, which contains infinity, and $$\rho$$ be a weight function on $$E$$. The main result of the paper provides the explicit strong asymptotic formula for the monic orthogonal polynomials $$Q_n(z,\sigma)$$ for the measure $$\sigma=\rho(t)|dt|+\alpha$$, where $$\rho$$ satisfies the Szegő condition, and $$\alpha$$ is a discrete measure with masses $$\lambda_j>0$$ at the points $$z_1,\ldots,z_n$$ in $$\Omega$$. The key ingredient is a certain extremal problem in a class of multivalued functions.

### MSC:

 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 30E15 Asymptotic representations in the complex plane
Full Text:

### References:

 [1] A. I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Mat. Sb. (N.S.) 125(167) (1984), no. 2, 231 – 258 (Russian). [2] A. Vol$$^{\prime}$$berg, F. Pekherstorfer, and P. Yuditskiĭ, Asymptotics of orthogonal polynomials in the case not covered by Szegő’s theorem, Funktsional. Anal. i Prilozhen. 40 (2006), no. 4, 22 – 32, 111 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 40 (2006), no. 4, 264 – 272. · Zbl 1116.33010 [3] Геометрическая теория функций комплексного переменного, Сецонд едитион. Едитед бы В. И. Смирнов. Щитх а супплемент бы Н. А. Лебедев, Г. В. Кузмина анд Ју. Е. Аленицын, Издат. ”Наука”, Мосцощ, 1966 (Руссиан). Г. М. Голузин, Геометриц тхеоры оф фунцтионс оф а цомплеш вариабле, Транслатионс оф Матхематицал Монограпхс, Вол. 26, Америцан Матхематицал Социеты, Провиденце, Р.И., 1969. · Zbl 0148.30603 [4] A. A. Gončar, The convergence of Padé approximants for certain classes of meromorphic functions, Mat. Sb. (N.S.) 97(139) (1975), no. 4(8), 607 – 629, 634 (Russian). [5] B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11 – 80 (Russian). With an appendix by I. M. Krichever. · Zbl 0478.58038 [6] E. M. Nikishin, The discrete Sturm-Liouville operator and some problems of function theory, Trudy Sem. Petrovsk. 10 (1984), 3 – 77, 237 (Russian, with English summary). · Zbl 0573.34023 [7] E. A. Rahmanov, The asymptotic behavior of the ratio of orthogonal polynomials, Mat. Sb. (N.S.) 103(145) (1977), no. 2, 237 – 252, 319 (Russian). [8] E. A. Rahmanov, The convergence of diagonal Padé approximants, Mat. Sb. (N.S.) 104(146) (1977), no. 2(10), 271 – 291, 335 (Russian). [9] M. Bello Hernández and G. López Lagomasino, Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theory 92 (1998), no. 2, 216 – 244. · Zbl 0897.42016 [10] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), no. 11, 1335 – 1425. , https://doi.org/10.1002/(SICI)1097-0312(199911)52:113.0.CO;2-1 · Zbl 0944.42013 [11] Valeri A. Kaliaguine, A note on the asymptotics of orthogonal polynomials on a complex arc: the case of a measure with a discrete part, J. Approx. Theory 80 (1995), no. 1, 138 – 145. · Zbl 0831.42015 [12] V. Kaliaguine and R. Benzine, Sur la formule asymptotique des polynômes orthogonaux associés à une mesure concentrée sur un contour plus une partie discrète finie, Bull. Soc. Math. Belg. Sér. B 41 (1989), no. 1, 29 – 46 (French, with English summary). · Zbl 0683.42027 [13] V. A. Kaliaguine and A. A. Kononova, Strong asymptotics for polynomials orthogonal on a system of complex arcs and curves: Szegő condition on and a mass points off the system, Publ. ANO-410, Univ. Sci. Techn. Lille, 2000. [14] R. Khaldi and F. Aggoune, On extremal polynomials on a system of curves and arcs, J. Eng. Appl. Sci. 2 (2007), no. 2, 372-376. · Zbl 1174.42335 [15] A. B. J. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche, and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1], Adv. Math. 188 (2004), no. 2, 337 – 398. · Zbl 1082.42017 [16] X. Li and K. Pan, Asymptotic behavior of orthogonal polynomials corresponding to measure with discrete part off the unit circle, J. Approx. Theory 79 (1994), no. 1, 54 – 71. · Zbl 0805.42017 [17] F. Marcellán and P. Maroni, Sur l’adjonction d’une masse de Dirac à une forme régulière et semi-classique, Ann. Mat. Pura Appl. (4) 162 (1992), 1 – 22 (French, with English and French summaries). · Zbl 0771.33008 [18] Franz Peherstorfer and Peter Yuditskii, Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3213 – 3220. · Zbl 0976.42012 [19] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. · JFM 65.0278.03 [20] Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. · Zbl 0791.33009 [21] Harold Widom, Extremal polynomials associated with a system of curves in the complex plane, Advances in Math. 3 (1969), 127 – 232. · Zbl 0183.07503
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.