Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic formulas for Toeplitz determinants


Author: Estelle Basor
Journal: Trans. Amer. Math. Soc. 239 (1978), 33-65
MSC: Primary 47B35; Secondary 42A56
MathSciNet review: 0493480
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The object of this paper is to find an asymptotic formula for determinants of finite dimensional Toeplitz operators generated by a class of functions with singularities. The formula is a generalization of the Strong Szegö Limit Theorem.


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

  • [1] E. W. Barnes, The theory of the G-function, Quart. J. Pure Appl. Math. 31 (1900), 264-313.
  • [2] H. Bateman, Higher transcendental functions, Vol. 1, Bateman Manuscript Project (A. Erdélyi, Editor), McGraw-Hill, New York, 1953. MR 15, 419.
  • [3] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695 (15,868a)
  • [4] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627 (16,914f)
  • [5] Herbert Buchholz, The confluent hypergeometric function with special emphasis on its applications, Translated from the German by H. Lichtblau and K. Wetzel. Springer Tracts in Natural Philosophy, Vol. 15, Springer-Verlag New York Inc., New York, 1969. MR 0240343 (39 #1692)
  • [6] M. E. Fisher and R. E. Hartwig, Toeplitz determinants. Some applications, theorems and conjectures, Adv. Chem. Phys. 15 (1968), 333-353.
  • [7] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space; English transl., Transl. Math. Monographs, vol. 18, Amer. Math. Soc., Providence, R. I., 1968. MR 39 #7447.
  • [8] Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR 0094840 (20 #1349)
  • [9] I. I. Hirschman Jr., On a formula of Kac and Achiezer, J. Math. Mech. 16 (1966), 167–196. MR 0208279 (34 #8089)
  • [10] I. I. Hirschman Jr., On a theorem of Szegö, Kac, and Baxter, J. Analyse Math. 14 (1965), 225–234. MR 0177250 (31 #1513)
  • [11] I. I. Hirschman Jr., Recent developments in the theory of finite Toeplitz operators, Advances in probability and related topics, Vol. 1, Dekker, New York, 1971, pp. 103–167. MR 0305130 (46 #4260)
  • [12] A. Lenard, Some remarks on large Toeplitz determinants, Pacific J. Math. 42 (1972), 137–145. MR 0331106 (48 #9440)
  • [13] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, rev. ed., Providence, R. I., 1959. MR 21 #5029.
  • [14] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469 (97k:01072)
  • [15] Harold Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II, Advances in Math. 21 (1976), no. 1, 1–29. MR 0409512 (53 #13266b)
  • [16] Harold Widom, Toeplitz determinants with singular generating functions, Amer. J. Math. 95 (1973), 333–383. MR 0331107 (48 #9441)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B35, 42A56

Retrieve articles in all journals with MSC: 47B35, 42A56


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0493480-X
PII: S 0002-9947(1978)0493480-X
Keywords: Asymptotic formula, Toeplitz determinant, singular generating function
Article copyright: © Copyright 1978 American Mathematical Society