Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system A n x = b [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from A n . The convergence features of these PCG have been naturally studied by means of the Weierstrass-Jackson Theorem [17,36,49], owing to the profound relationship between the spectral features of the matrices A n , generated by the Fourier coefficients of a continuous function f , and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f . On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense. Subject Classification (1991): 65F10, 47B25, 41A36, 15A18
Mathematics