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Book Review

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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 Rolanía, B. de la Calle Ysern, and G. López Lagomasino, Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures, J. Approx. Theory 139 (2006), no. 1-2, 223–256. MR 2220040,
  • 2. K. M. Case, Orthogonal polynomials revisited, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York, 1975, pp. 289–304. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. MR 0390322
  • 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. It⋅s, and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2, 395–430. MR 1174420
  • 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, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 557–560 (Russian). MR 0043315
  • 7. Leonid Golinskii, On the scientific legacy of Ya. L. Geronimus (to the hundredth anniversary), Self-similar systems (Dubna, 1998) Joint Inst. Nuclear Res., Dubna, 1999, pp. 273–281. 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. Paul R. Halmos, I have a photographic memory, American Mathematical Society, Providence, RI, 1987. MR 934204
  • 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. Michael T. Lacey, Carleson’s theorem: proof, complements, variations, Publ. Mat. 48 (2004), no. 2, 251–307. MR 2091007,
  • 12. G. G. Lorentz, Mathematics and politics in the Soviet Union from 1928 to 1953, J. Approx. Theory 116 (2002), no. 2, 169–223. MR 1911079,
  • 13. Attila Máté and Paul G. Nevai, Bernstein’s inequality in 𝐿^{𝑝} for 0<𝑝<1 and (𝐶,1) bounds for orthogonal polynomials, Ann. of Math. (2) 111 (1980), no. 1, 145–154. MR 558399,
  • 14. Math Hist,
  • 15. H. N. Mhaskar, Introduction to the theory of weighted polynomial approximation, Series in Approximations and Decompositions, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1469222
  • 16. Paul Nevai, Letter to a friend, J. Approx. Theory 46 (1986), no. 1, 3–8. Papers dedicated to the memory of Géza Freud. MR 835719,
  • 17. Math Intelligence Test,
  • 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 pdf.
  • 20. George Pólya, The Pólya picture album: encounters of a mathematician, Birkhäuser Boston, Inc., Boston, MA, 1987. Edited and with an introduction and a biography by G. L. Alexanderson. MR 898436
  • 21. Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088
  • 22. Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR 2105089
  • 23. B. Simon, Orthogonal Polynomials on the Unit Circle, the website,
  • 24. Barry Simon, OPUC on one foot, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 4, 431–460. MR 2163705,
  • 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
  • 26. Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517
  • 27. Gábor Szegő, Collected papers. Vol. 1, Contemporary Mathematicians, Birkhäuser, Boston, Mass., 1982. 1915–1927; Edited by Richard Askey; Including commentaries and reviews by George Pólya, P. C. Rosenbloom, Askey, L. E. Payne, T. Kailath and Barry M. McCoy. MR 674482
  • 28. Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789
  • 29. Vilmos Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283–303. MR 1785285,
  • 30. Vilmos Totik, Orthogonal polynomials, Surv. Approx. Theory 1 (2005), 70–125. MR 2221567
  • 31. David S. Watkins, Some perspectives on the eigenvalue problem, SIAM Rev. 35 (1993), no. 3, 430–471. MR 1234638,

Review Information:

Reviewer: Paul Nevai
Affiliation: The Ohio State University
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