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

Abstract

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

Korovkin tests, approximation, and ergodic theory

“…Incidentally this fact establishes that the sequence of spaces {M(U n )} is asymptotically close in the distribution sense to the sequence of linear spaces { f ∈L1 A n (f )}, and this fact translates to fast convergence when P N (n) (A n (f )) is used as preconditioner for A n (f ) in pcg-like methods [6,20,22,10].…”

“…In this section we recall some results [20,10] of approximation theory based on the concepts of linear positive operators and in the spirit of the Korovkin theorems [17].…”

“…On the other hand, the convergence in L 1 is valid for the Cesàro (or Fejér) operators [8] which, indeed, can be vieved as a special case of the operators {L n [U ](·)} when the spaces {M n = M(U n )} are the circulants [20] associated to the sequence {U = U n } of the Fourier transforms. Other nonclassical examples [20,22,10] of such sequences of operators are obtained by using other spaces, such as those based on sine, cosine, or Hartley transforms [16,20].…”

“…Indeed, as a special case of the general result, we consider also the optimal approximation of the multilevel Toeplitz matrices with s × t unstructured blocks by algebras/matrix spaces: we obtain that the optimal approximation also distributes as f for all the known trigonometric algebras [10,16]. This approximation result has two aspects: the first is related to the use of cg-like algorithms [5,21,20,31]. The second is not so classical, because we propose an algorithm for approximating a multivariate Lebesgue integrable function f .…”

“…We stress that we use the matrix version of the Korovkin theorems developed in [20,22] and the simple but powerful technique used by Tilli in the Toeplitz context [29]. Indeed, as a special case of the general result, we consider also the optimal approximation of the multilevel Toeplitz matrices with s × t unstructured blocks by algebras/matrix spaces: we obtain that the optimal approximation also distributes as f for all the known trigonometric algebras [10,16].…”