zbMATH — the first resource for mathematics

On Szegö’s asymptotic formula for Toeplitz determinants and generalizations. (English) Zbl 0661.30001
A new proof was given for the asymptotic formula for the Toeplitz \(\det er\min ant:\)
D\({}_{n-1}(f)=\exp (nc_ 0+\sum^{\infty}_{k=1}kc_ kc_{- k}+o(1))\) as \(n\to \infty\) \((c_ k\) are complex Fourier coefficients of log f), proved earlier by G. Szegö [Meddel. Lunds Univ. Mat. Sem. Suppl.-Band M. Riesz, 228-238 (1952; Zbl 0048.042)]. The formula was proved in the first part for complex valued f under the assumption \(\sum^{\infty}_{-\infty}| k| | c_ k|^ 2\), using the well-known integral representation of a Toeplitz determinant as a basis. In the second part an asymptotic formula for Hankel determinants was \(proved:\)
D\({}_{n-1}(f)D_{n-1}(1)^{-1}=\exp (na_ 0+(1/8)\sum^{\infty}_{k=1}ka^ 2_ k+o(1))\) as \(n\to \infty\), \(a_ k\) are the coefficients in the expansion of f in Chebyshev polynomials. Finally a similar formula was proved for a determinant being a function on \(\gamma\) (\(\gamma\) is a Jordan curve in the complex plane).
Reviewer: V.Burjan

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators