A short survey on preconditioners and Korovkin-type theorems (Q2047302)

This is a survey paper summarising how Korovkin-type theorems can be applied to construct preconditioners to solve linear systems \(Ax=b\). A typical Korovkin theorem says that for a sequence \(L_n\) of positive linear operators it holds in a particular sense that \(\lim_{n\to\infty} L_n(f)=f\) for all \(f\) in some function class if \(\lim_{n\to\infty}L_n(h)=h\) for \(h\) in some finite test set. These theorems have been generalized in different ways to Korovkin type theorems and were applied to preconditioners for Toeplitz matrices. Let \(P_{U_n}\) be the projection on \(M_{U_n}\), with \(M_{U_n}\) the algebra of Hermitian matrices that are \(U_n\)-similar to a diagonal matrix. That is \(P_{U_n}(A)\) are the matrices in \(M_{U_n}\), closest (in Frobenius norm) to the Hermitian matrix \(A\in\mathbb{C}^{n\times n}\). If \(A(f)\) is a Toeplitz operator with symbol \(f\) and \(A_n(f)\) its \(n\times n\) leading submatrix, then a possible Korovkin type theorem says that \(P_{U_n}(A_n(f))-A_n(f)\) converges to 0 for \(f\) in the \(C^*\) algebra generated by \(\{g_1,\ldots,g_m\}\) of real \(2\pi\) periodic functions if the convergence holds for the test set \(\{g_1,\ldots,g_m,\sum_{k=1}^m g_k^2\}\). In the context of preconditioners the matrices \(U_n\) are usually (generalized) Vandermonde matrices evaluating trigonometric functions in equidistant grid points of modulus 1 to generate circulant, Hartley, or \(\tau\)-algebras. Past and current research is surveyed. All the definitions are given, including the particular types of convergence to be considered. For the proofs the reader is most often referred to the literature. Generalizations include infinite operators and the non-Hermitian matrix case, and more general function spaces for \(f\). Some suggestions for future research are made. The paper is quite readable although there are some typographical problems with norm sign and \(i\) is used as an index and as \(\sqrt{-1}\) in te same formula.