Asymptotic behavior of Toeplitz matrices and determinants

Abstract

We consider the inverse X _N and determinant D_N(c) of an N×N Toeplitz matrix C_N=[c_i−j] ^N−1₀ as N ar∞. Under the condition that there exists a monotonic decreasing summable bound b _n ≧|c _n |+|c _−n |, and that the generating function $c(\theta ) = \sum\limits_{n = - \infty }^\infty {c_n e^{i{\text{ }}n{\text{ }}\theta } }$ does not vanish, we construct a matrix iterative process which yields (i) explicit asymptotic formulae for the elements of X_N when v(c) = (2π)⁻¹ [arg{c(2π)}−arg{c(0)}] is zero. Thence we obtain (ii) expressions for the constants, and bounds on the remainder, in the asymptotic formula

$$\ln D_N (c) = N{\text{ }}k_0 (c) + E_0 (c) + E_{1,N} (c) + \mathcal{R}_N (c),$$

and (iii) the extension of this formula to the case of general integral v(c). Under certain further conditions the monotonicity of E_1,N+ℛ_N is proved. We discuss various identities for D_N which apply when c(θ) is a rational function of e^iθ and mention a conjecture for D _N when c(θ) has zeros, and is discontinuous with arbitrary v(c).