Toeplitz matrices with an exponential growth of entries and the first Szego limit theorem

The Toeplitz (or block Toeplitz) matrices S(r) = {s(j-k)}(k, j = 1)(r), generated by the Taylor coefficients at zero of analytic functions phi(lambda) = s(0)/2 + Sigma(p = 1)(infinity) s (-P)lambda(p) and psi(mu) = s(0)/2 + Sigma(p = 1)(infinity) s(P) mu(p), are considered. A method is proposed for removing the poles of phi and psi or, in other words, for replacing S(infinity), whose entries grow exponentially, by a matrix (S) over cap(infinity) = {(s) over cap(j-k)}(k, j = 1)(infinity) with better behaviour and the same asymptotics of <(Delta)over cap>(r) = det (S) over cap(r) (r --> infinity) as the sequence Delta(r) = der S(r). A Szego-type limit formula for the case when S(r) = S(r)* (r greater than or equal to n(0)) have a fixed number of negative eigenvalues is obtained. (C) 2000 Academic Press.