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Toeplitz determinants with one Fisher-Hartwig singularity. (English) Zbl 0909.47019
The authors study the asymptotic behaviour of the Toeplitz determinants \(T_n (c)\) as \(n\) tends to infinity in the case of symbols with one singularity. This symbol has a form \[ c(e^{i\theta} e^{i\theta}) \prod_{r}^R t_{\beta _r}\left( e^{i(\theta-i\theta _r)}\right) u_{\alpha _r}\left( e^{i(\theta-i\theta _r)}\right) \] where \(t_\beta \left (e^{i\theta} \right) ^{i\beta(\theta - \pi)}\), \(u_{\alpha}\left(e^{i\theta} \right)( 2- 2 \cos \theta)^\alpha\) and \(b\) are sufficiently smooth nonvanishing functions.

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A15 Determinants, permanents, traces, other special matrix functions
Full Text: DOI
[1] Basor, E.L., Asymptotic formulas for Toeplitz determinants, Trans. amer. math. soc., 239, 33-65, (1978) · Zbl 0409.47018
[2] Basor, E.L.; Tracy, C.A., The fisher – hartwig conjecture and generalizations, Phys. A, 177, 167-173, (1991)
[3] Basor, E.L.; Morrison, K.E., The fisher – hartwig conjecture and Toeplitz eigenvalues, Linear algebra appl., 202, 129-142, (1994) · Zbl 0805.15004
[4] Basor, E.L.; Morrison, K.E., The extended fisher – hartwig conjecture for symbols with multiple jump discontinuities, Toeplitz operators and related topics: the harold Widom anniversary volume, Operator theory: advances and applications, 71, (1994), Birkhäuser Verlag Basel/Boston/Berlin, p. 16-28 · Zbl 0818.47017
[5] Böttcher, A.; Silbermann, B., The asymptotic behavior of Toeplitz determinants for generating functions with zeros of integral order, Math. nachr., 102, 79-105, (1981) · Zbl 0479.47025
[6] Böttcher, A.; Silbermann, B., Toeplitz matrices and determinants with fisher – hartwig symbols, J. funct. anal., 63, 178-214, (1985) · Zbl 0592.47016
[7] Böttcher, A.; Silbermann, B., Toeplitz operators and determinants generated by symbols with one fisher – hartwig singularity, Math. nachr., 127, 95-124, (1986) · Zbl 0613.47024
[8] Böttcher, A.; Silbermann, B., Analysis of Toeplitz operators, (1989), Akademie Verlag Berlin · Zbl 0666.47016
[9] Fisher, M.E.; Hartwig, R.E., Toeplitz determinants: some applications, theorems, and conjectures, Adv. chem. phys., 15, 333-353, (1968)
[10] Forrester, P.J.; Pisani, C., The hole probability in log-gas and random matrix systems, Nucl. phys. B, 374, 720-740, (1992) · Zbl 0992.82504
[11] Khavin, V.P.; Nikolskij, N.K., Commutative harmonic analysis I, Encyclopaedia of mathematical sciences, 15, (1991), Springer-Verlag Berlin/New York
[12] Libby, R., Asymptotics of determinants and eigenvalue distribution for Toeplitz matrices associated with certain discontinuous symbols, (1990), Univ. of California Santa Cruz
[13] McCoy, B.M.; Wu, T.T., The two-dimensional Ising model, (1973), Harvard Univ. Press Cambridge
[14] Simon, B., Notes on infinite determinants of Hilbert space operators, Adv. math., 24, 244-273, (1977) · Zbl 0353.47008
[15] G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function, Festskrift Marcel Riesz, Lund, 1952, 222, 238
[16] Whittaker, E.T.; Watson, G.N., A course of modern analysis, (1952), Cambridge Univ. Press London/New York · Zbl 0108.26903
[17] Widom, H., Toeplitz determinants with singular generating functions, Amer. J. math., 95, 333-383, (1973) · Zbl 0275.45006
[18] Widom, H., Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanishing index, Topics in operator theory: ernst D. Hellinger memorial volume, Operator theory: advances and applications, 48, pp., (1990), Birkhäuser Verlag Basel/Boston/Berlin, p. 387-425
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