A-Statistical extension of the Korovkin type approximation theorem

In this paper, using the concept ofA-statistical convergence which is a regular (non-matrix) summability method, we obtain a general Korovkin type approximation theorem which concerns the problem of approximating a functionf by means of a sequenceL _n f of positive linear operators.     

Korovkin type approximation theorem via Ka−convergence on weighted spaces

In this paper, we study the Korovkin type approximation theorem for K_a‐ convergence, which is an interesting convergence method on weighted spaces. We also study the rate of K_a?convergence by using the weighted modulus of continuity and afterwards, we present a nontrivial application.     

$$K_{a}$$-convergence and Korovkin type approximation

In the present paper, we study a Korovkin type approximation theorem in the setting of \(K_{a}\)-convergence that contains the classical result. We also study the rate of \(K_{a}\)-convergence and afterwards, we give some concluding remarks.     

On a discrete Korovkin theorem

In [G. A. Anastassiou, A discrete Korovkin theorem, J. Approx. Theory 45 (1985), pp. 383–388, Theorem 3], a discrete Korovkin theorem was given. We restate the theorem here and its proof, correcting a mistake in the above reference.     

A discrete Korovkin theorem

In this paper we give a sufficient condition for the pointwise Korovkin property on B(X), the space of bounded real valued functions on an arbitrary countable set X = {x_l,…, x_j,…}. Our theorem follows from its L_p(X, μ) analogue (and conversely); here 1 p < ∞ and μ is a positive finite measure on X such that μ({x_j}) > 0 for all j.     

Sequential type Korovkin theorem on -test functions

Let be a sequence of bounded linear operators on such that and for every . It is proved that for every .

     

Korovkin type approximation theorems on the space of continuously differentiable fnctions

It is studied Korovkin type approximation theorems on C⁽¹⁾ ([0, 1]) the space of continuously differentiable functions on the unit interval. It is proved that test functions for which Korovkin type approximation theorems hold depending on norms of C⁽¹⁾ ([0, 1]).     

Korovkin type approximation theorem for BBH type operators via <Emphasis Type="Italic">J</Emphasis>-convergence

This paper provides a Korovkin type approximation theorem for a class of positive linear operators including Bleimann-Butzer and Hahn operators via J-convergence.     

Korovkin tests, approximation, and ergodic theory

We consider sequences of matrices with a block structure spectrally distributed as an -variate matrix-valued function , and, for any , we suppose that is a linear and positive operator. For every fixed we approximate the matrix in a suitable linear space of matrices by minimizing the Frobenius norm of when ranges over . The minimizer is denoted by . 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 is distributed as ,

2.

the sequence is distributed as the constant function (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 summability factor theorem for absolute summability involving quasi-monotone sequences

In this paper we prove a theorem on _|A|k,k≥1${\left|A\right|}_k,k\ge 1$, summability factors for an infinite series by replacing a weighted mean matrix with a triangular matrix.