Toeplitz Matrices with an Exponential Growth of Entries and the First Szegِ Limit Theorem

The Toeplitz (or block Toeplitz) matrices S(r)={sj−k}rk, j=1, generated by the Taylor coefficients at zero of analytic functions ϕ(λ)=s02+∑∞p=1 s−pλp and ψ(μ)=s02+∑∞p=1 spμp, are considered. A method is proposed for removing the poles of ϕ and ψ or, in other words, for replacing S(∞), whose entries grow exponentially, by a matrix Ŝ(∞)={ŝj−k}∞k, j=1 with better behaviour and the same asymptotics of Δ(r)=det Ŝ(r) (r→∞) as the sequence Δr=det S(r). A Szegö-type limit formula for the case when S(r)=S(r)* (r⩾n0) have a fixed number of negative eigenvalues is obtained.