Rate of convergence in Szegő's asymptotic formula for Toeplitz determinants (Q1586346)

Given a function \(f\in L_1[0, 2\pi]\) with the Fourier coefficients \(c_k\), \(k=0, \pm 1,\dots\), the Toeplitz determinants are defined by \(D_{n-1}(f)=\det (c_{k-l})_{k,l=0}^{n-1}\). Let \(\Omega \) denote the class of all functions \(f\in L_1[0, 2\pi]\) satisfying \(\sum_{-\infty}^{\infty} k|c_k|^2<\infty\). For such a function \textit{K. Johansson} [Bull. Sci. Math., II. Sér. 112, No.~3, 257-304 (1988; Zbl 0661.30001)] gave the following generalization of the Szegő asymptotic formula: \(|\log D_{n-1}(\exp(f)) -nc_0-\sum_{k=1}^{\infty}kc_kc_{-k}|\to 0\) as \(n\to\infty\). Moreover, if a real-valued function \(f\) and its conjugate function \(\tilde f\) belong to the class \(C^1\), then \(|\log D_{n-1}(\exp(f))-nc_0-\sum_{k=1}^{\infty}kc_kc_{-k}|\leq{c(f)\over n}\), \ \(n=1, 2,\dots\) . Using the concept of fractional derivatives and the \(L_p\) moduli of smoothness of fractional orders, the author obtains the same rate of convergence for a broader class of functions, namely for all real-valued functions \(f\in \Omega \) having the conjugate trigonometric function \(\tilde f\in L_{\infty}[0, 2\pi]\).