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Book Information:

Author: Barry Simon
Title: Orthogonal polynomials on the unit circle, Parts 1 and 2
Additional book information: American Mathematical Society, Providence, RI, pt. 1, 2005, xxvi+466 pp., ISBN 978-0-8218-3446-6, US$89.00; pt. 2, xxii+578 pp., ISBN 978-0-8218-3675-0, US$99.00; ISBN 978-0-8218-3757-3, US$149.00, set

References [Enhancements On Off] (What's this?)

  • 1. D. Barrios, B. de la Calle, and G. López Lagomasino, Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov type measures, J. Approx. Theory, 139 (2006) (volume dedicated to Barry Simon), 223-256. MR 2220040 (2006m:42002)
  • 2. K. M. Case, Orthogonal polynomials revisited, in ``Theory and Application of Special Functions'', R. A. Askey (Editor), Proceedings of an advanced seminar sponsored by the Mathematics Research Center, University of Wisconsin-Madison, March 21-April 2, 1975. Academic Press, New York, 1975, pp. 289-304. MR 0390322 (52:11148)
  • 3. M. Fekete and J. von Neumann, Über die Lage der Nullstellen gewisser Minimumpolynome, Jahresbericht der Deutschen Mathematiker-Vereinigung, 31 (1922), 128-138.
  • 4. A. S. Fokas, A. R. Its, and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys., 147 (1992), 395-430. MR 1174420 (93h:81115)
  • 5. C. F. Gauss, Methodus nova integralium valores per approximationem inveniendi, Commentationes Societatis regiae scientarium Gottingensis recentiores, 3 (1814), 39-76.
  • 6. I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation by its spectral function. (Russian), Doklady Akad. Nauk SSSR (N.S.), 77 (1951), 557-560. MR 0043315 (13:240f)
  • 7. L. Golinskii, On the scientific legacy of Ya. L. Geronimus (to the hundredth anniversary), in ``Self-Similar Systems'', V. B. Priezzhev and V. P. Spiridonov (Editors), Proceedings of the International Workshop held in Dubna, July 30-August 7, 1998, Joint Inst. Nuclear Res., Dubna, 1999, pp. 273-281, and http://theor.jinr.ru/meetings/98/sss/proceedings.html. MR 1819440
  • 8. L. Golinskii and V. Totik, Orthogonal Polynomials: From Jacobi to Simon, in Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday, P. Deift, F. Gesztesy, P. Perry, and W. Schlag (eds.), Proceedings of Symposia in Pure Mathematics, vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 821-874.
  • 9. P. Halmos, I Have a Photographic Memory, American Mathematical Society, Providence, RI, 1987. MR 934204 (89f:01067)
  • 10. C. G. J. Jacobi, Über Gauss' neue Methode, die Werthe der Integrale näherungsweise zu finden, Crelle J. Reine Angew. Math., 1 (1826), 301-308.
  • 11. M. T. Lacey, Carleson's theorem: proof, complements, variations, Publ. Mat., 48 (2004), no. 2, 251-307, and http://www.arxiv.org/abs/math.CA/0307008. MR 2091007 (2005e:42020)
  • 12. G. G. Lorentz, Mathematics and politics in the Soviet Union from 1928 to 1953, J. Approx. Theory, 116 (2002), 169-223, and http://www.math.ohio-state.edu/AT/LORENTZ/JAT02-0001_final.pdf. MR 1911079 (2003c:01059)
  • 13. A. Máté and P. Nevai, Bernstein's inequality in $ L^p$ for $ 0<p<1$ and $ (\CC,1)$ bounds for orthogonal polynomials, Ann. Math., 111 (1980), 145-154. MR 558399 (81c:42003)
  • 14. Math Hist, http://www-history.mcs.st-andrews.ac.uk/.
  • 15. H. N. Mhaskar, Introduction to the theory of weighted polynomial approximation, Series in Approximations and Decompositions, Vol. 7, World Scientific, 1996. MR 1469222 (98i:41014)
  • 16. P. Nevai, Letter to a friend, J. Approx. Theory, 46 (1986), 4-8. MR 835719 (87j:01062a)
  • 17. Math Intelligence Test, http://www.math.ohio-state.edu/~nevai/QUIZ/quiz.txt.
  • 18. A. B. J. Novikoff, Special System of Orthogonal Polynomials, Ph.D. Dissertation, Stanford University, 1954.
  • 19. I. V. Ostrovskii, Kharkov Mathematical Society, in ``European Mathematical Society'', Newsletter No. 34, December, 1999, pp. 26-27, and http://emis.kaist.ac.kr/newsletter/newsletter34. pdf.
  • 20. G. Pólya, The Pólya Picture Album: Encounters of a Mathematician, G. L. Alexanderson (Editor), Birkhäuser, 1987. MR 898436 (88i:01125)
  • 21. B. Simon, Orthogonal Polynomials on the Unit Circle, V.1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005. MR 2105088 (2006a:42002a)
  • 22. B. Simon, Orthogonal Polynomials on the Unit Circle, V.2: Spectral Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005. MR 2105089 (2006a:42002b)
  • 23. B. Simon, Orthogonal Polynomials on the Unit Circle, the website, http://www.math.caltech.edu/opuc.html.
  • 24. B. Simon, OPUC on one foot, Bull. Amer. Math. Soc. (N.S.), 42 (2005), no. 4, 431-460, and http://www.ams.org/bull/2005-42-04/S0273-0979-05-01075-X/S0273-0979-05-01075-X. pdf. MR 2163705 (2006e:42039)
  • 25. B. Simon, CMV matrices: Five years after, in Proceedings of the W. D. Evans' 65th Birthday Conference, J. Comput. Appl. Math., XXX (2007), to appear, and http://arxiv.org/abs/math.SP/0603093.
  • 26. G. Szego, Orthogonal Polynomials, AMS Colloquium Series, Vol. 23, American Mathematical Society, Providence, R.I., 1939; 4th edition, 1975. MR 0372517 (51:8724)
  • 27. G. Szego, Collected papers, Vol. 1, 1915-1927, edited by R. Askey, Birkhäuser, 1982. MR 674482 (84d:01082a)
  • 28. V. Totik, Weighted approximation with varying weight, Springer Lecture Notes, Vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789 (96f:41002)
  • 29. V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math., 81 (2000), 283-303. MR 1785285 (2001j:42021)
  • 30. V. Totik, Orthogonal Polynomials, Surveys in Approximation Theory, online journal at http://www.math.technion.ac.il/, 1 (2005), 70-125, and http://www.math.technion.ac.il/sat/papers/3/. MR 2221567 (2007a:42055)
  • 31. David S. Watkins, Some perspectives on the eigenvalue problem, SIAM Review, 35 (1993), 430-471. MR 1234638 (94h:65039)

Review Information:

Reviewer: Paul Nevai
Affiliation: The Ohio State University
Email: paul@nevai.us
Journal: Bull. Amer. Math. Soc. 44 (2007), 447-470
MSC (2000): Primary 05E35; Secondary 00A17
Published electronically: April 10, 2007
Review copyright: © Copyright 2007 American Mathematical Society
American Mathematical Society