Positivity 11 (2007), 387–398
2007 Birkh¨auser Verlag Basel/Switzerland
Matrix Summability and Positive Linear
Ozlem G. Atlihan and Cihan Orhan
Abstract. In the present paper we prove a Korovkin type approximation the-
orem for a sequence of positive linear operators acting from a weighted space
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.
Mathematics Subject Classiﬁcation (2000). 41A25; 41A36; 47B38.
Keywords. Matrix summability, sequence of positive linear operators, weight
function, weighted space, Korovkin type theorem, rates of convergence.
Most of the classical approximation operators tend to converge to the value of
function being approximated. However, at points of discontinuity, they often con-
verge to the average of the left and right limits of the function. There are, however,
some exceptions that do not converge at points of simple discontinuity . In this
case the matrix summability methods of Ces´aro type are strong enough to correct
the lack of convergence . The purpose this paper is to study a Korovkin type
approximation of a function f by means of a sequence of positive linear operators
from a weighted space C
into a weighted space B
with the use of a matrix
summability method which includes both convergence and almost convergence.
Approximation theory has important applications in the theory of polyno-
mial approximation, in functional analysis, numerical solutions of diﬀerential and
integral equations (, , ).
Now we recall the concepts of weight functions and weighted spaces consid-
ered in , . Let R denotes the set of real numbers. The function ρ is called
a weight function if it is continuous on R and
ρ(x)=∞ and ρ(x) ≥ 1 (for all x ∈ R). (1)