Toeplitz operators on weighted \(\ell_ p\) spaces and associated asymptotics
The author in his paper, is concerned with finite Toeplitz matrices generated, in some way, by a piecewise continuous function. More specifically we can find in it asymptotic information about the eigenvalues of such finite matrices. The first instance is Theorem 3.3 which shows that the finite Toeplitz matrices \[ T_ n[\Phi]=(\Phi_{j-k})_{j,k=0,...,n}, \] where \(\Phi\) is a piecewise continuous complex valued map on \(S^ 1\), and \(\lambda_ k\) are the eigenvalues of the image of \(\Phi\). Therefore \(f(T_ n[\Phi])\) makes sense for a large class of suitably defined analytic functions f and an asymptotic formula for \[ tr f(T_ n[\Phi])=\sum f(\lambda_ i) \] is given, where the \(\lambda_ i's\) are the eigenvalues of \(T_ n[\Phi]\). This formula is a generalization of a Szegö limit theorem [J. Math. Analysis Appl. 120, No.-1, 25-38 (1986; Zbl 0609.47036) or \textit{H. Widom}, Adv. Math. 21, 1-29 (1976; Zbl 0344.47016)]. The paper has some kind of bright so that it is easy to read it.
finite Toeplitz matrices generated by a piecewise continuous function
asymptotic information about the eigenvalues
scientific article; zbMATH DE number 4173717