On the Singular Values of Generalized Toeplitz Matrices

For a bounded function σ defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [0, 1]\times \mathbb{T} $$\end{document}, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{s^{(n)}_{k}\}^{n+1}_{k=1} $$\end{document} be the set of singular values of the (n + 1) x (n + 1) matrix whose (j, k)-entries are equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{2\pi}\int^{2\pi}_{0} \sigma(\frac{k}{n},e^{i\theta})e^{-i(j-k)\theta}d\theta,\qquad j,k = 0, 1,...,n. $$\end{document} These matrices can be thought of as variable-coefficient Toeplitz matrices or generalized Toeplitz matrices. Matrices of the above form can be also thought of as the discrete analogue of pseudodifferential operators. Under a certain smoothness assumption on the function σ, we prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum_{k=1}^{n+1} f(s^{2}_{k})\quad = \quad c_{1}\cdot (n+1) + c_{2} + o(1)\qquad as \enskip n \rightarrow \infty, $$\end{document} where the constant c1 and a part of c2 are shown to have explicit integral representations. The other part of c2 turns out to have a resemblance to the Toeplitz case. This asymptotic formula can be viewed as a generalization of the classical theory on singular values of Toeplitz matrices.