A Korovkin-type theory for finite Toeplitz operators via matrix algebras (Q1293999)

The approximation of finite selfadjoint Toeplitz operators \(A_n(\cdot)\) by means of matrix algebra operators is considered. The Hermitian Toeplitz matrix \(A_n(f)\) is generated by a Lebesgue-integrable real-valued function \(f\) defined in \([-\pi,\pi]\) in the sense that the entries of \(A_n(f)\) along the \(k-\)th diagonal are given by the \(k-\)th Fourier coefficient of \(f\). For solving the Toeplitz system \(A_nx=b\) Frobenius-optimal preconditioners are chosen in some matrix algebras and are defined by minimizing the Frobenius distance from \(A_n\). It is shown that the optimal preconditioners can be used to define linear positive operators uniformly approximating the function \(f\). By modifying the Korovkin theorem it is provided a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms.