The asymptotic behaviour of
Toeplitz determinants Dn(f), as n →∞, is considered for nonnegative generating
functions f(𝜃) with a finite number of isolated zeros 𝜃ν, in the neighborhood of which
f(𝜃) ∼|ei𝜃− ei𝜃γ|αγ where αν> 0. Using an argument suggested by Szegö, an
upper bound of the form Dn(f) < C ⋅ Gn+1(n + 1)σ is derived, where G is the
geometrical mean of f and σ = 1∕4∑αν2. 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 Dn, as n →∞, for some
special f’s. It is conjectured that this asymptotic equality is generally true for the
class of f’s considered.
