Korovkin theorems and linear positive Gram matrix algebra approximations of Toeplitz matrices

In this paper we are concerned with the approximation of Toeplitz matrices generated by continuous 2 pi-periodic functions f : I --> R, with I = [-pi, pi]. For this purpose we, define a class of matrix algebras, related to suitable choices of Gram functions, where we look for good preconditioners. In particular, we construct these preconditioners through linear operators and linear positive operators (LPOs) approximating in some sense the function f. Then, by making use of some matrix Versions [39, 42, 41] of the Korovkin and Weierstrass theorems, we analyze the convergence features of old and new preconditioners. Finally, among the given results are adapted in order to deal with L-1 generating functions and the related preconditioners are compared with the ones devised by using band-Toeplitz matrices [6, 36]. (C) 1998 Elsevier Science Inc. All rights reserved.