The Toeplitz (or block Toeplitz) matrices S(r) = {s(j-k)}(k, j = 1)(r), gen
erated 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 + S
igma(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(infin
ity), 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 sam
e 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 c
ase 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.