zbMATH — the first resource for mathematics

Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. (English) Zbl 0976.42012
Szegő theory gives the strong asymptotics of orthogonal polynomials on the unit circle under the condition that the absolutely continuous part of the measure has an integrable logarithm (Szegő’s condition). This result can then easily be transformed to give strong asymptotics of orthogonal polynomials on an interval (usually \([-1,1]\), but the authors consider \([-2,2]\)). The authors investigate orthogonal polynomials with respect to a positive measure \(\mu\), for which the absolutely continuous part satisfies Szegő’s condition on the interval \([-2,2]\), but in addition they allow \(\mu\) to have a discrete part outside \([-2,2]\), concentrated at mass points \(x_k\). They assume a Blaschke condition \(\sum \sqrt{x_k^2-4} < \infty\) and prove the strong asymptotics for the orthonormal polynomials and their leading coefficients. The result involves the Szegő function for the absolutely continuous part and a Blaschke product of the points \(\zeta_k=(x_k-\sqrt{x_k^2-4})/2\) in the open unit disk.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30D50 Blaschke products, etc. (MSC2000)
Full Text: DOI
[1] J. S. Geronimo and K. M. Case, Scattering theory and polynomials orthogonal on the real line, Trans. Amer. Math. Soc. 258 (1980), no. 2, 467 – 494. · Zbl 0436.42018
[2] Ya.L. Geronimus, Polynomials orthogonal on a circle and interval, Pergamon Press, New York, 1960. · Zbl 0093.26503
[3] A.A. Gonchar, On convergence of Padé approximants for certain classes of meromorphic functions, Mat.Sb. 97 (1975), English translation: Math. USSR-Sb. 26.
[4] Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. · Zbl 0405.33009
[5] Paul Nevai, Research problems in orthogonal polynomials, Approximation theory VI, Vol. II (College Station, TX, 1989) Academic Press, Boston, MA, 1989, pp. 449 – 489. · Zbl 0725.33001
[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. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, Translations of Mathematical Monographs, vol. 92, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas. · Zbl 0733.41001
[8] F. Peherstorfer and P Yuditskii, Asymptotic behavior of polynomials orthonormal on a homogeneous set, manuscript. · Zbl 1032.42028
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.