Vol. 42, No. 1, 1972

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Some remarks on large Toeplitz determinants

Andrew Lenard

Vol. 42 (1972), No. 1, 137–145

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 = 14 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.

Mathematical Subject Classification 2000
Primary: 47B35
Received: 8 March 1971
Published: 1 July 1972
Andrew Lenard