Korovkin tests, approximation, and ergodic theory

Abstract

Abstract. We consider sequences of s · k(n) × t · k(n) matrices {An(f )} with a block structure spectrally distributed as an L 1 p-variate s × t matrix-valued function f , and, for any n, we suppose that An(·) is a linear and positive operator. For every fixed n we approximate the matrix An(f ) in a suitable linear space Mn of s · k(n) × t · k(n) matrices by minimizing the Frobenius norm of An(f ) − Xn when Xn ranges over Mn. The minimizerXn is denoted by P k(n) (An(f )). We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts:1. the sequence {P k(n) (An(f ))} is distributed as f , 2. the sequence {An(f )−P k(n) (An(f ))} is distributed as the constant function 0 (i.e. is spectrally clustered at zero). The first result is an ergodic one which can be used for solving numerical approximation theory problems. The second has a natural interpretation in the theory of the preconditioning associated to cg-like algorithms.

A unifying approach to abstract matrix algebra preconditioning

“…This check is really trivial because A n (p) is tridiagonal or pentadiagonal and λ j is the (j, j) entry of the matrix U * n A n (p)U n that can be explicitly computed for the three test polynomials. We recall that the same powerful statement (10) with rank(∆ n ) = o(n) and with f being merely integrable has been recently proved in [38].…”

“…On the other hand, the functional way is more abstract and more powerful at least in the case of the optimal Frobenius approximation for which we can use the matrix version of the Korovkin theorem [35,38].…”

The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

“…We now complement the previous theorem with a short discussion regarding the hypothesis {C −1 n T n [f ]} n ∼ σ 1. Going back to the analysis in [3,16], we have the following picture:…”

“…B) when C n is the Frobenius optimal preconditioner for T n [f ], the key assumption {C + n T n [f ]} n ∼ σ 1 holds if f is sparsely vanishing and simply Lebesgue integrable (such a general result was proven quite elegantly by combining the Korovkin theory [16] and the GLT analysis [5]); C) By combining item A) and item B), we can update Theorem 3.5, by including the case where C n is not necessarily invertible. It is enough to replace C −1 n by C + n , taking into account that the assumption of f sparsely vanishing will imply the presence of at most o(n) zero eigenvalues both in the matrix C n and in the preconditioned matrix C + n T n [f ].…”

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

“…), it is interesting to observe that in [55,56] these LPOs have been used in connection with Theorem 5.5 in order to solve some nontrivial approximation problems (e.g. rational approximation of f/g with g ≥ 0 and f/g continuous, "construction" of the essential range of f/g with g ≥ 0 and f , g ∈ L 2 , etc.)…”