Abstract
Szegő’s First Limit Theorem provides the limiting statistical distribution of the eigenvalues of large Toeplitz matrices. Szegő’s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the First and Second Limit Theorems to Kac–Murdock–Szegő (KMS) matrices. These are matrices whose entries along the diagonals are not necessarily constants, but modeled by functions. We clarify and extend some existing results, and explain some apparently contradictory results in the literature.
Similar content being viewed by others
References
Agnew A.F., Bourget A.: The semiclassical density of states for the quantum asymmetric top. J. Phys. A, 41, 185205 (2008)
Agnew A.F., Bourget A.: A trace formula for a family of Jacobi operators. Anal. Appl. (Singap.), 7, 115–130 (2009)
Avram F.: On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Related Fields, 79, 37–45 (1988)
R. Bhatia, Matrix Analysis, Grad. Texts in Math., 169, Springer-Verlag, 1997.
Borcea J.B., Shapiro B.Z.: Root asymptotics of spectral polynomials for the Lamé operator. Comm. Math. Phys., 282, 323–337 (2008)
A.Böttcher and S.M. Grudsky, Spectral Properties of Banded Toeplitz Matrices, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005.
Böttcher A., Silbermann B.: Toeplitz matrices and determinants with Fisher–Hartwig symbols. J. Funct. Anal., 63, 178–214 (1985)
Bourget A.: New identities for the spectrum of the quantum Euler top. J. Phys. A, 43, 265201 (2010)
Bourget A.: Spectral density of Jacobi matrices with small deviations. Constr. Approx., 36, 375–398 (2012)
Bourget A., McMillen T.: Spectral inequalities for the quantum asymmetric top. J. Phys. A, 42, 095209 (2009)
A. Bourget and T. McMillen, Asymptotics of determinants of discrete Schrödinger operators, to appear in J. Spect. Theor., arXiv:1609.04125.
Bourget A., McMillen T.: A first Szegő’s limit theorem for a class of non-Toeplitz matrices. Constr. Approx., 45, 47–63 (2017)
Brown A., Halmos P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math., 213, 89–102 (1964)
Bump D., Diaconis P., Hicks A., Miclo L., Widom H.: An exercise(?) in Fourier analysis on the Heisenberg group. Ann. Fac. Sci. Toulouse Math. (6), 26, 263–288 (2017)
D. Bump, P. Diaconis, A. Hicks, L. Miclo and H. Widom, Useful bounds on the extreme eigenvalues and vectors of matrices for Harper’s operators, In: Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., 259, Birkhäuser, 2017, pp. 235–265.
Deift P., Its A., Krasovsky I.: Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher–Hartwig singularities. Ann. of Math. (2), 174, 1243–1299 (2011)
P. Deift and K.D.T.R. McLaughlin, A Continuum Limit of the Toda Lattice, Mem. Amer. Math. Soc., 131, no. 624, Amer. Math. Soc., Providence, RI, 1998.
Donatelli M., Mazza M., Serra-Capizzano S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys., 307, 262–279 (2016)
Ehrhardt T., Shao B.: Asymptotic behavior of variable-coefficient Toeplitz determinants. J. Fourier Anal. Appl., 7, 71–92 (2001)
Garoni C., Serra-Capizzano S.: The theory of locally Toeplitz sequences: a review, an extension, and a few representative applications. Bol. Soc. Mat. Mex., 22, 529–565 (2016)
U. Grenander and G. Szegő, Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences, Univ. of California Press, Berkeley–Los Angeles, 1958.
Hirschman I.I. Jr.: The spectra of certain Toeplitz matrices. Illinois J. Math., 11, 145–159 (1967)
Kac M.: Asymptotic behaviour of a class of determinants. Enseignement Math. (2), 15, 177–183 (1969)
Kac M.: On certain Toeplitz-like matrices and their relation to the problem of lattice vibrations. J. Stat. Phys., 151, 785–795 (2013)
Kac M., Murdock W.L., Szegő G.: On the eigenvalues of certain Hermitian forms. J. Rational Mech. Anal., 2, 767–800 (1953)
Kuijlaars A.B.J., Serra-Capizzano S.: Asymptotic zero distribution of orthogonal polynomials with discontinuously varying recurrence coefficients. J. Approx. Theory, 113, 142–155 (2001)
Kuijlaars A.B.J., Van Assche W.: The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory, 99, 167–197 (1999)
McMillen T.: On the eigenvalues of double band matrices. Linear Algebra Appl., 431, 1890–1897 (2009)
McMillen T., Bourget A., Agnew A.: On the zeros of complex Van Vleck polynomials. J. Comput. Appl. Math., 223, 862–871 (2009)
Mejlbo L.C., Schmidt P.F.: On the determinants of certain Toeplitz matrices. Bull. Amer. Math. Soc., 67, 159–162 (1961)
Mejlbo L.C., Schmidt P.F.: On the eigenvalues of generalized Toeplitz matrices. Math. Scand., 10, 5–16 (1962)
Noschese S., Reichel L.: The structured distance to normality of banded Toeplitz matrices. BIT, 49, 629–640 (2009)
Noschese S., Reichel L.: The structured distance to normality of Toeplitz matrices with application to preconditioning. Numer. Linear Algebra Appl., 18, 429–447 (2011)
Parter S.V.: On the distribution of the singular values of Toeplitz matrices. Linear Algebra Appl., 80, 115–130 (1986)
Schmidt P., Spitzer F.: The Toeplitz matrices of an arbitrary Laurent polynomial. Math. Scand., 8, 15–38 (1960)
Serra-Capizzano S.: A note on the asymptotic spectra of finite difference discretizations of second order elliptic partial differential equations. Asian J. Math., 4, 499–514 (2000)
S. Serra-Capizzano, Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000), Linear Algebra Appl., 366 (2003), 371–402.
Serra-Capizzano S.: The GLT class as a generalized Fourier analysis and applications.. Linear Algebra Appl., 419, 180–233 (2006)
Shao B.: A trace formula for variable-coefficient Toeplitz matrices with symbols of bounded variation. J. Math. Anal. Appl., 222, 505–546 (1998)
Shao B.: A trace formula for a class of variable-coefficient block Toeplitz matrices. Integral Equations Operator Theory, 45, 359–374 (2003)
Shao B.: On the singular values of generalized Toeplitz matrices. Integral Equations Operator Theory, 49, 239–254 (2004)
Shapiro B., Takemura K., Tater M.: On spectral polynomials of the Heun equation. II. Comm. Math. Phys., 311, 277–300 (2012)
Tilli P.: Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl., 278, 91–120 (1998)
P. Tilli, Asymptotic spectral distribution of Toeplitz-related matrices, In: Fast Reliable Algorithms for Matrices with Structure, SIAM, Philadelphia, PA, 1999, pp. 153–187.
Tilli P.: Some results on complex Toeplitz eigenvalues. J. Math. Anal. Appl., 239, 390–401 (1999)
Trefethen L.N., Chapman S.J.: Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math., 57, 1233–1264 (2004)
L.N. Trefethen and M. Embree, Spectra and Pseudospectra. The Behaviors of Nonnormal Matrices and Operators, Princeton Univ. Press, Princeton, NJ, 2005.
Trench W.F.: Spectral distribution of generalized Kac–Murdock–Szegő matrices. Linear Algebra Appl., 347, 251–273 (2002)
Trotter H.F.: Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegő. Adv. in Math., 54, 67–82 (1984)
Tyrtyshnikov E.E.: A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl., 232, 1–43 (1996)
Ullman J.L.: A problem of Schmidt and Spitzer. Bull. Amer. Math. Soc., 73, 883–885 (1967)
Widom H.: Asymptotic behavior of block Toeplitz matrices and determinants. Advances in Math., 13, 284–322 (1974)
Widom H.: Asymptotic behavior of block Toeplitz matrices and determinants. II. Advances in Math., 21, 1–29 (1976)
H. Widom, Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanishing index, In: Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl., 48, Birkhäuser, 1990, pp. 387–421.
H. Widom, Eigenvalue distribution for nonselfadjoint Toeplitz matrices, In: Toeplitz Operators and Related Topics, Santa Cruz, CA, 1992, Oper. Theory Adv. Appl., 71, Birkhäuser, 1994, pp. 1–8.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Toshiyuki Kobayashi
About this article
Cite this article
Bourget, A., Loya, A.A. & McMillen, T. Spectral asymptotics for Kac–Murdock–Szegő matrices. Jpn. J. Math. 13, 67–107 (2018). https://doi.org/10.1007/s11537-018-1640-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11537-018-1640-2