Asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices (Q2748467)

The authors study the asymptotics of the determinants of matrices of the form \(M_n(\phi)=T_n(\phi)+H_n(\phi)\), where \(T_n(\phi)=(\phi_{j-k})_{j,k=0}^{n-1}\) is a Toeplitz matrix and \(H_n(\phi)=(\phi_{j+k+1})_{j,k=0}^{n-1}\) is a Hankel matrix; here \(\phi_k=[\phi]_k\) is the \(k\)th Fourier coefficient of the generating function \(\phi\). The authors assume that \(\phi\) is a piecewise continuous function of the form NEWLINE\[NEWLINE \phi(e^{i\theta})= b(e^{i\theta})t_{\beta_+}(e^{i\theta})t_{\beta_-}(e^{i(\theta-\pi)}) \prod_{r=1}^R t_{\beta_r}(e^{i(\theta-\theta_r)}), NEWLINE\]NEWLINE where \(t_\beta(e^{i\theta})=e^{i\beta(\theta-\pi)}\), \(0<\theta<2\pi\), and \(b\) is a sufficiently smooth nonvanishing function on the unit circle with winding number zero. Under some assumptions on the parameters \(\beta_\pm, \beta_1,\dots,\beta_R\) and \(\theta_1,\dots,\theta_R\) and on the smoothness of \(b\), the authors prove the asymptotic formula NEWLINE\[NEWLINE \det M_n(\phi)\sim (\exp[\log b]_0)^n n^{\Omega_M}E_M \quad (n\to\infty), NEWLINE\]NEWLINE where \(\Omega_M\) and \(E_M\) are completely identified constants depending on \(\varphi\).NEWLINENEWLINEThe authors assume that piecewise continuous generating functions do not possess a jump at both \(e^{i\theta}\) and \(e^{-i\theta}\). Unfortunately, this extra assumption rules out all even functions. Notice that the authors were able to improve the results of the present paper and to remove this extra assumption in a subsequent paper [Oper. Theory, Adv. Appl. 135, 61--90 (2002; Zbl 1025.15011)].