Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices

Abstract

We compute the asymptotics of the determinants of certain n×n Toeplitz + Hankel matrices T n (a) + H n (b) as n → ∞ with symbols of Fisher-Hartwig type. More specifically we consider the case where a has zeros and poles and where b is related to a in specific ways. Previous results of Deift, Its and Krasovsky dealt with the case where a is even. We are generalizing this in a mild way to certain non-even symbols.E if a is a sufficiently well-behaved function. Such a result is an analogue of the classical Szegö-Widom limit theorem [17] for Toeplitz determinants.The above determinant is a special case of more general determinants, det (a j−k + b j+k+1 ) j,k=0,...,n−1,

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

“…Although there are no results in the literature for the Toeplitz+Hankel determinants where w is supported on the line, the case where w is supported on the unit circle has been considered under specific assumptions. E. Basor and T. Ehrhardt have studied different aspects of these determinants in a series of papers [2,3,4,5,6] via operator-theoretic tools over the last 20 years or so. In [13], the Riemann-Hilbert technique which has already been proven very effective to study the asymptotics of Toeplitz and Hankel determinants was extended for the first time to the determinants of Toeplitz+Hankel matrices generated by the same symbol w = φ, where the Hankel weight is supported on T. In that work the symbol was assumed to be of Fisher-Hartwig type and it was further required that the symbol be even, i.e., w =w.…”

“…In [13], the Riemann-Hilbert technique which has already been proven very effective to study the asymptotics of Toeplitz and Hankel determinants was extended for the first time to the determinants of Toeplitz+Hankel matrices generated by the same symbol w = φ, where the Hankel weight is supported on T. In that work the symbol was assumed to be of Fisher-Hartwig type and it was further required that the symbol be even, i.e., w =w. 1 In [2], by employing the relevant results in [13], the authors managed for the first time to find the asymptotics of Toeplitz+Hankel determinants for certain non-coinciding symbols. Indeed, they considered φ(z) = c(z)φ 0 (z), and w(z) = c(z)d(z)w 0 (z), (1.4) where the functions c and d are assumed to be smooth and nonvanishing on the unit circle with zero winding number.…”

“…Indeed, the characteristic polynomial det(H n [w] − λI) of the Hankel matrix H n [w] is a particular Toeplitz+Hankel determinant, with φ(z) ≡ −λ. Clearly in the case of characteristic polynomial of a Hankel determinant, there is no relationship between φ and w, so to study the asymptotics of this determinant, one can not refer to the works [13] or [2] mentioned above. Here again we are directed to a methodological issue which has to be addressed at a fundamental level by formulation of a suitable Riemann-Hilbert problem.…”

“…We have been able to solve the model problem for the class of symbols (1.4) considered in [2], in the absence of Fisher-Hartwig singularities. It is important to discuss the relevancy of the two conditions assumed to be satisfied by the function d in [2], in our Riemann-Hilbert framework (see (1.4) and below).…”

On determinants identity minus Hankel matrix

“…The problem which we study here fits into the more general framework of asymptotics of determinants of Hankel, Toeplitz and Hankel plus Toeplitz matrices. These are well‐studied objects, see, for example, and references therein. Exhaustive answers to various questions related to the asymptotics of Toeplitz and Hankel determinants have been found, however the behaviour of completely general Hankel plus Toeplitz determinants is not entirely understood yet.…”