A short survey on preconditioners and Korovkin-type theorems

Abstract

Korovkin-type theorems in the commutative as well as noncommutative set up have many applications in various fields. In this survey, we discuss the applications of noncommutative Korovkin-type theorems in numerical linear algebra. Consider the equation $$Ax=b$$ where A is an $$n \times n$$ matrix with Toeplitz structure. In 1988, Tony Chan invented the optimal circulant preconditioner C for the system and observed that they are useful in getting faster convergence for iteration processes associated with classical splittings and with (preconditioned) Krylov methods. In the same spirit, several researchers have considered this problem to get efficient preconditioners such as Hartley, $$\epsilon$$ -circulant etc. In 1999, Stefano Serra-Capizzano generalized this technique by considering an arbitrary sequence of matrix algebras $$\{M_{U_n}\}$$ that includes algebras associated with fast transforms like Fourier, Trigonometric, Hartley, Wavelet etc. He showed that in this general situation, a sequence of matrices $$\{P_{U_n}(A_n)\}$$ in $$M_{U_n}$$ is a preconditioner to the Toeplitz matrix sequence $$\{A_n\}$$ if $$P_{U_n}(A_n)-A_n$$ converges to the 0 matrix in the sense of the singular value clustering or eigenvalue clustering, if all the involved matrices are Hermitian. In 2013, these notions were generalized into the setting of operators acting on infinite dimensional Hilbert spaces. The connection of preconditioners with the noncommutative Korovkin-type theorems is the most interesting aspect of this study. The classical theorem of Korovkin unified several approximation processes such as Bernstein, Weierstrass, Fejer etc. Similarly, the Korovkin-type theorems obtained in the context of matrix/operator sequences here unify several preconditioning techniques such as Fourier, Hartley etc. These results find applications in spectral approximation problem for bounded self-adjoint operators. The most recent interests include Korovkin-type theorems for nonnormal Toeplitz operators on various function spaces. A short survey of all these developments are presented in this article.