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 | References | Similar Articles | Additional Information

Abstract: Let be a positive Borel measure in [-1,1] with a.e. It is shown that the polynomials orthonormal with respect to this measure oscillate almost everywhere in [-1,1]. A function is also described that is a pointwise bound for , 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0818456-1

Keywords:
Orthogonal polynomials,
Szegö's theory,
weak convergence,
mean convergence

Article copyright:
© Copyright 1986
American Mathematical Society