Relative modular uniform approximation by means of the power series method with applications

Kuldip Raj,Anu Choudhary

doi:10.33044/revuma.v60n1a11

Abstract

We introduce the notion of relative convergence by means of a four dimensional matrix in the sense of the power series method, which includes Abel's as well as Borel's methods, to prove a Korovkin type approximation theorem by using the test functions {1,y,z,y2+z2} and a double sequence of positive linear operators defined on modular spaces. We also endeavor to examine some applications related to this new type of approximation.

Highlights

Introduction and preliminariesKorovkin type approximation plays an important role in summability theory
The classical Bohman–Korovkin theorem establishes the uniform convergence in the space C[a, b] of all continuous real valued functions defined on the interval [a, b], for a sequence of positive linear operators acting on C[a, b] assuming the convergence only on the test functions 1, x, and x2
From the existing literature we found that there is no comparison between statistical A-approximation process and the power series method

Summary

So the given double sequence is convergent to

1 3 in the sense of the power series method but it is not convergent in the Pringsheim sense. A double sequence {fi,j} of functions whose terms belong to Lρ(H2) is said to be relatively modularly convergent to a function f ∈ Lρ(H2) in the sense of the power series method iff there exists a scale function σ(y, z) ∈ X(H2), |σ(y, z)| = 0, such that qi,j cidj ρ i,j=0. Let S = {Si,j} be a sequence of positive linear operators from G into X(H2), XS ⊂ G containing C∞(H2), A = (aklij) a non-negative RH-regular summability matrix method, and σ(y, z) ∈ X(H2) an unbounded function with σ(y, z) = 0 such that lim sup c,d→R−. Let S = {Si,j} be a double sequence of positive linear operators from G into X(H2) satisfying (1.4), A = (aklij) be a non-negative RH-regular summability matrix method, ρ a strongly finite, monotone, absolutely continuous, Q quasi-convex modular on X(H2), and σm(y, z) an unbounded function satisfying.

This implies

This implies that

Observe that