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What is beyond Szegö's theory of orthogonal polynomials?

  • Orthogonal Polynomials
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Rational Approximation and Interpolation

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1105))

Abstract

Consider a system {φn} of polynomials orthonormal on the unit circle with respect to the measure dµ with μ′ > 0 almost everywhere. Then

$$\mathop {\lim }\limits_{n \to \infty } \int_{ - \pi }^\pi {\left| {\left| {\phi _n \left( {e^{i\theta } } \right)} \right|\sqrt {\mu '\left( \theta \right)} - 1} \right|^2 } d\theta = 0.$$

.

This result enables one to extend many results of Szegö's theory to the case μ′ > 0.

This material is based upon work supported by the National Science Foundation under Grant Nos. MCS 8100673 (first author) and MCS-83-00882 (second author) and by the PSC-CUNY Research Award Program of the City University of New York under Grant No. 662043 (first author). The third author made his contributions to the paper while visiting the Ohio State University.

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics, Brooklyn College of the City University of New York, 11210, Brooklyn, New York, USA

    Attila Máté

  2. Department of Mathematics, Ohio State University, 43210, Columbus, Ohio, USA

    Paul Nevai

  3. Bolyai Institute, University of Szeged, 6720, Szeged, Hungary

    Vilmos Totik

Authors
  1. Attila Máté
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  2. Paul Nevai
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  3. Vilmos Totik
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Editor information

Peter Russell Graves-Morris Edward B. Saff Richard S. Varga

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© 1984 Springer-Verlag

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Máté, A., Nevai, P., Totik, V. (1984). What is beyond Szegö's theory of orthogonal polynomials?. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072436

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  • DOI: https://doi.org/10.1007/BFb0072436

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13899-0

  • Online ISBN: 978-3-540-39113-5

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