Matrix Summability and Positive Linear Operators

Abstract

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space Cρ 1 into a weighted space Bρ 2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.

Approximation theorems by positive linear operators in weighted spaces

“…When we study the A-statistical analogue of Theorem 1.1, we shall need the matrix A be regular and { D j C ω 1 →B ω 2 } be uniform boundedness. However, we also can use the weaker conditions (3.16) and (3.18) to get an A-summable analogue of Theorem 1.1 without the regularity of A or the uniform boundedness of { D j C ω 1 →B ω 2 } (see [2]). …”

“…The one dimensional cases were proved in [2] and [8], respectively. Of course, all results of this paper are also valid for higher dimensional cases.…”

Korovkin‐Type Theorems in Weighted Lp‐Spaces via Summation Process

“…Replacing the ordinary convergence by 𝒜 -summability some approximation results have been studied in [ 11 – 13 ] and in the special cases [ 14 , 15 ]. Also, Korovkin-type theorems in weighted space via 𝒜 -summability have been studied in [ 16 , 17 ].…”

A-summation process and Korovkin-type approximation theorem for double sequences of positive linear operators

“…The main purpose of using summability theory has always been to make a nonconvergent sequence converge. Some results regarding matrix summability for positive linear operators may be found in the paper [1], [2], [14].…”