Oscillatory behavior of orthogonal polynomials

Máté, Attila; Nevai, Paul; Totik, Vilmos

doi:10.1090/S0002-9939-1986-0818456-1

Oscillatory behavior of orthogonal polynomials

Authors: Attila Máté, Paul Nevai and Vilmos Totik
Journal: Proc. Amer. Math. Soc. 96 (1986), 261-268
MSC: Primary 42C05
DOI: https://doi.org/10.1090/S0002-9939-1986-0818456-1
MathSciNet review: 818456
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Abstract: Let $d\alpha$ be a positive Borel measure in [-1,1] with $\alpha ' > 0$ a.e. It is shown that the polynomials ${p_n}$ orthonormal with respect to this measure oscillate almost everywhere in [-1,1]. A function is also described that is a pointwise bound for ${p_n}$ , exceeded only on sets of small measure. It is shown that is the best possible.

[1] G. Freud, Orthogonal polynomials, Pergamon Press, New York, 1971.
[2] Ja. L. Geronimus, Orthogonal polynomials, Two Papers on Special Functions, Amer. Math. Soc. Transl. (2) 108 (1977), 37-130. (The Russian original appeared as an appendix added to the Russian Translation of [14], GIFML, Moscow, 1962.)
[3] P. R. Halmos, Measure theory, 2nd printing, Springer-Verlag, New York and Berlin, 1974.
[4] A. Máté and P. Nevai, Remarks on E. A. Rahmanov's paper "On the asymptotics of the ratio of orthogonal polynomials", J. Approx. Theory 36 (1982), 64-72. MR 673857 (83m:42017)
[5] A. Máté, P. Nevai and V. Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constructive Approx. 1 (1985), 63-69. MR 766095 (85j:42045)
[6] -, Strong and weak convergence of orthogonal polynomials, manuscript.
[7] -, Necessary conditions for the weighted mean convergence of Fourier series in orthogonal polynomials, J. Approx. Theory (to appear). MR 840398 (87j:42074)
[8] P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. No. 213 (1979). MR 519926 (80k:42025)
[9] -, Orthogonal polynomials defined by a recurrence relation, Trans. Amer. Math. Soc. 250 (1979), 369-384. MR 530062 (80d:42011)
[10] -, On orthogonal polynomials, J. Approx. Theory 25 (1979), 34-37.
[11] E. A. Rahmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR-Sb. 32 (1977), 199-213. (Russian original: Mat. Sb. 103 (1977), 237-252.) MR 0445212 (56:3556)
[12] -, On the asymptotics of the ratio of orthogonal polynomials. II, Math. USSR-Sb. 46 (1983), 105-117. (Russian original: Mat. Sb. 118 (1982), 104-117.)
[13] G. Pólya and G. Szegö, Problems and theorems in analysis. I, Springer-Verlag, New York and Berlin, 1972.
[14] G. Szegö, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.