Abstract
Two important techniques to achieve the Jackson type estimation by Kantorovich type positive linear operators in spaces are introduced in the present paper, and three typical applications are given.
Highlights
It is well known that Kantorovich type operators are usually used for approximation in Lp spaces
We find out that the Jackson order in Lp spaces to approximate f(x) ∈ Lp[0,1] by the operators in (3) or (5) is decided completely by the kernels {Kn,j(x)}nj=1, or by the kernel function Qn(x, t)
The positive linear operators used in Lp approximation can be classified according to the properties of their kernels
Summary
It is well known that Kantorovich type operators are usually used for approximation in Lp spaces. This paper discusses how well the function f(x) ∈ Lp[0,1] can be approximated by the discrete Kantorovich-type operators such as n (j+1)/(n+1). We find out that the Jackson order in Lp spaces to approximate f(x) ∈ Lp[0,1] by the operators in (3) or (5) is decided completely by the kernels {Kn,j(x)}nj=1, or by the kernel function Qn(x, t). On applying this idea, we need only to investigate the properties of the kernels to obtain the magnitude of the Jackson order of the corresponding operators, which seems to be a different approach from the past Lp approximating methods
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