×

A strong Spitzer-Stone theorem for a Toeplitz matrix with singular symbol defined by a class of analytic functions. (Un théoreme de Spitzer-Stone fort pour une matrice de Toeplitz à symbole singulier défini par une classe de fonctions analytiques.) (French. English summary) Zbl 1145.47024

Let \(T(f)\) be a Toeplitz operator with symbol \(f\) and \(T_N(f)\in{\mathbb C}^{(N+1)\times(N+1)}\) denote the Toeplitz matrix whose elements are the Fourier coefficients of \(f\). The purpose of this paper is to give estimates for the elements \(b_{kl}\) of the inverse of \(T_N(f)\) as \(N\) tends to infinity. Two cases are considered: \(f(z)=| f_1(z)| ^2\) (regular case) or \(f(z)=| 1-z| ^2| f_1(z)| ^2\) (singular case), where \(f_1\) is a function holomorphic in a disc \(| z|\). In the singular case, this improves the Spitzer–Stone result [cf.F.L.Spitzer and C.J.Stone, Ill.J.Math.4, 253–277 (1960; Zbl 0124.34403)] in that (1) it also includes second and even third order terms, (2) that the estimate which holds for finite indices \((k,l)\) also holds for indices behaving like integer values of \((Nx,Ny)\) with \(0\leq x,y<1\), (3) the error term has the form \(O(1/\rho^N)\) with \(1<\rho<R\). More precise results are obtained for the special case where \(f_1\) is a rational function. Also, expressions for the trace and the sum of the elements of the inverse are given. \(R\) with \(R>1\).
In the singular case, this improves the Spitzer–Stone result [cf.F.L.Spitzer and C.J.Stone, Ill.J.Math.4, 253–277 (1960; Zbl 0124.34403)] in that (1) it also includes second and even third order terms, (2) that the estimate which holds for finite indices \((k,l)\) also holds for indices behaving like integer values of \((Nx,Ny)\) with \(0\leq x,y<1\), (3) the error term has the form \(O(1/\rho^N)\) with \(1<\rho<R\). More precise results are obtained for the special case where \(f_1\) is a rational function. Also, expressions for the trace and the sum of the elements of the inverse are given.
For the regular case, classical prediction theory has shown that the predictor polynomial (essentially the orthogonal Szegő polynomial) approximates the inverse \(1/f_1\).
All approximants depend on the Fourier coefficients of \(1/f_1\), on \(R\) (the region of holomorphy), and in the singular case also on the values of \(f_1\) and \(1/f_1\) and their derivatives in \(z=1\) (the singular point).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A09 Theory of matrix inversion and generalized inverses
65F05 Direct numerical methods for linear systems and matrix inversion
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Citations:

Zbl 0124.34403
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Adamyan (V.M.).— Asymptotic properties for positive and Toeplitz matrices. Operator theory : Adv. and Appl. 43, p. 17-38 (1990). · Zbl 0701.47012
[2] Basor (E. L.).— Asymptotic formulas for Toeplitz determinants. Trans. Amer. Math. Soc., 239, p. 33-65 (1978). · Zbl 0409.47018
[3] Bleher (P. M.).— Inversion of Toeplitz matrices. Trans. Moscow Math. Soc. 2, p. 201Ð-224 (1981). · Zbl 0473.15004
[4] Böttcher (A.).— The constants in the asymptotic formulas by Rambour and Seghier for the inverse of Toeplitz matrices. 99, p. 43-45 (2004). · Zbl 1070.47015
[5] Böttcher (A.), Silbermann (B.).— Toeplitz matrices and determinants with Fisher-Hartwig symbols. J. Funct. Anal. 63, p. 178-214 (1985). · Zbl 0592.47016
[6] Böttcher (A.), Silbermann (B.).— Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity. Math. Nachr. 127, p. 95-124 (1986). · Zbl 0613.47024
[7] Böttcher (A.), Silbermann (B.).— Analysis of Toeplitz operators, Springer Verlag (1990). · Zbl 0732.47029
[8] Böttcher (A.), Widom (H.).— From Toeplitz eigenvalues through Greens kernels to higherorder Wirtinger-Sobolev inequalities. arXiv, math.FA/0412269 v1 (2004).
[9] Böttcher (A.), Widom (H.).— Two elementary derivations of the pure Fisher-Hartwig determinant (2004) to appear. · Zbl 1081.47033
[10] Coursol (J.), Dacunha-Castelle (D.).— Remarques sur l’approximation de la vraisemblance d’un processus gaussien stationnaire. Teor. Veroyatnost. i Primenen. 27(1), p. 155-160 (1982). · Zbl 0511.60036
[11] Dow (M.).— Explicit inverses of Toeplitz and associated matrices. Anziam J. 44, p. 185-215 (2003). · Zbl 1116.15300
[12] Ehrhardt (T.).— Toeplitz determinants with several Fisher-Hartwig singularities. Dissertation, Technische Universität Chemnitz, 1997. · Zbl 0910.47020
[13] Ehrhardt (T.).— A status report on the asymptotic behaviour of Toeplitz determinants with Fisher-Hartwig singularities. Oper. Theory Adv. Appl., p. 217-241 (2001). · Zbl 0993.47028
[14] Ehrhardt (T.), Silbermann (B.).— Toeplitz determinants with one Fisher-Hartwig singularity. Journal of Functional Analysis 148, p. 229-256 (1997). · Zbl 0909.47019
[15] Fisher (M. E.), Hartwig (R. E.).— Toeplitz determinants ; some applications, theorems, and conjectures. Adv. Chem. Phys. 15, p. 333-353 (1968).
[16] Grenander (U.), Szego (G.).— Toeplitz forms and their applications. Chelsea, New York, 2nd ed. edition (1984). · Zbl 0611.47018
[17] Ibrahimov (I.), Rozanov (Y.).— Processus al« eatoires gaussiens. Editions Mir de Moscou, 1 edition (1974). · Zbl 0291.60021
[18] Kesten (H.).— Random walk with absorbing barriers and Toeplitz forms. Illinois J. of Math. 5, p. 267-290 (1961). · Zbl 0129.30401
[19] Landau (H.J.).— Maximum entropy and the moment problem. Bulletin (New Series) of the american mathematical society. 16(1), p. 47-77 (1987). · Zbl 0617.42004
[20] Rambour (P.), Seghier (A.).— Exact and asymptotic inverse of the Toeplitz matrix with polynomial singular symbol. CRAS, 336, ser.1. p. 399-400 (2003). · Zbl 1012.65025
[21] Rambour (P.), Seghier (A.).— Inversion asymptotique des matrices de Toeplitz à symboles singuliers. Extension d’un résultat de H. Kesten. Prépublications de l’Université Paris-sud (2003).
[22] Rambour (P.), Seghier (A.).— Formulas for the inverses of Toeplitz matrices with polynomially singular symbols. Integr. equ. oper. theory. 50, p. 83-114 (2004). · Zbl 1069.47027
[23] Rambour (P.), Rinkel (J-M.).— Application to random walks of the exact inverse of the Toeplitz matrix with singular rational symbol. Probablity and Mathematical Statistics. 25, p. 183-195 (2005). · Zbl 1102.47062
[24] Rinkel (J-M.).— Inverses et propriétés spectrales des matrices de Toeplitz à symbole singulier, Thèse (2001).
[25] Sakhnovich (A.L.), Spitkovsky (I.M.).— Block-Toeplitz matrices and associated properties of a Gaussian model on the half axis. Teoret.Mat.Fiz.. 63, p. 154-160 (1985). · Zbl 0608.47026
[26] Seghier (A.).— Inversion asymptotique des matrices de toeplitz en d-dimension. J. of funct. analysis. 67, p. 380-412 (1986). · Zbl 0589.47023
[27] Seghier (A.).— Thèse de doctorat d’état. Université de Paris sud (1988).
[28] Spitzer (F. L.), Stone (C. J.).— A class of Toeplitz forms and their applications to probability theory. Illinois J. Math. 4, p. 253-277 (1960). · Zbl 0124.34403
[29] Vladimirov (V.S.), Volovich (I.V.).— A model of statistical physics. Teoret. Mat.Fiz. 54, p. 8-22 (1983).
[30] Widom (H.).— Extreme eigenvalues of N-dimensional convolution operators. Trans. Amer. Math. Soc. 106, p. 391-414 (1963). · Zbl 0205.14603
[31] Widom (H.).— Toeplitz determinant with singular generating function. Amer. J. Math. 95, (1973). · Zbl 0275.45006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.