An ergodic theorem for classes of preconditioned matrices

Korovkin tests, approximation, and ergodic theory

“…Therefore, for any positive M and for any positive (with the exception, at most, of a countable set of values of [21,33]), we find that…”

“…We suppose that the singular values of {A n (f )} are distributed as f with regard to the probability space (Ω, µ) [31,29,21] in the Weyl-Tyrtyshnikov sense: for any F ∈ C 0 (continuous function with bounded support) the ergodic formula…”

“…Therefore the composition P K (A n (f )) with K = k(n) is another linear positive operator in a somewhat different sense [21], because, under the assumption that s = t, it maps nonnegative definite matrix-valued functions into nonnegative definite matrices.…”

“…Moreover the optimal Frobenius approximation of matrices {A n (f )}, enjoying (4) and (6), is a way of defining new sequences {L n [U ](·)} of linear positive operators for which the Korovkin theorem applies in the L 1 case. We notice that the Bernstein polynomial operators [8] are linear and positive and satisfy the Korovkin test, but the approximation does not hold in L 1 [27] in the sense stated in equation (21). 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.…”

“…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 .…”