Some remarks on large Toeplitz determinants

The asymptotic behaviour of Toeplitz determinants D n (f), as n~+ oo, is considered for nonnegative generating functions f() with a finite number of isolated zeros v , in the neighborhood of which f() ~ I e -e i | *v where a v > 0. Using an argument suggested by Szeg, an upper bound of the form Dn(f) < C*G n+1 (n + l) is derived, where G is the geometrical mean of / and ^ = 1/4^ <*l* Using some identities in the theory of orthogonal polynomials, and specifically facts about Jacobi polynomials, it is shown that the above bound is actually asymptotically equal D ny as n -> oo, for some special /'s. It is conjectured that this asymptotic equality is generally true for the class of /'s considered.