In this paper, using the notion of A-statistical convergence from the summability theory, we obtain a Korovkin-type theorem for the approximation by means of matrixvalued linear positive operators. We show that our theorem is more applicable than the result introduced by S. Serra-Capizzano [A Korovkin based approximation of multilevel Toeplitz matrices (with rectangular unstructured blocks) via multilevel trigonometric matrix spaces, SIAM J. Numer. Anal., 36 (1999), 1831–1857]. Furthermore, we compute the A-statistical rates of our approximation.
In this study, using the concept of A-statistical convergence we investigate a Korovkin type approximation result for a sequence of positive linear operators defined on the space of all continuous real valued functions on any compact subset of the real m-dimensional space.
We obtain asymptotic formulas with arbitrary order of accuracy for the eigenvalues and eigenfunctions of a nonselfadjoint ordinary differential operator of order
whose coefficients are Lebesgue integrable on [0, 1] and the boundary conditions are strongly regular. The orders of asymptotic formulas are independent of smoothness of the coefficients.
In the present work, using the concept of
-statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued
-continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.