Abstract
Generalization of Szász-Mirakyan operators has been considered by Jain, 1972. Using these generalized operators, we introduce new sequences of positive linear operators which are the integral modification of the Jain operators having weight functions of some Beta basis function. Approximation properties, the rate of convergence, weighted approximation theorem, and better approximation are investigated for these new operators. At the end, we generalize Jain-Beta operator with three parameters α, β, and γ and discuss Voronovskaja asymptotic formula.
Highlights
For 0 < θ < ∞, |κ| < 1, let wκ (i, θ) = θ(θ +iκ)i−1 e−(θ+iκ) i!i = 0, 1, 2, . . . ; (1) ∞ ∑wκ (i, θ) = 1. (2)
We introduce new sequences of positive linear operators which are the integral modification of the Jain operators having weight functions of some Beta basis function
Dubey and Jain [9] considered a parameter γ in the definition of [8]. Motivated by such types of operators we introduce new sequence of linear operators as follows: For x ∈ [0, ∞) and γ > 0
Summary
In 1970, Jain [2] introduced and studied the following class of positive linear operators: Pnκ (f, x). In Gupta et al [8] they considered integral modification of the Szasz-Mirakyan operators by considering the weight function of Beta basis functions. Dubey and Jain [9] considered a parameter γ in the definition of [8] Motivated by such types of operators we introduce new sequence of linear operators as follows: For x ∈ [0, ∞) and γ > 0, Pnκ,γ (f, x) = ∑wκ (i, nx) ∫ bn,i,γ (t) f (t) dt + e−nxf (0) , i=1. The operators defined by (5) are the integral modification of the Jain operators having weight function of some Beta basis function. We propose Stancu-type generalization of (5) and discuss some local approximation properties and asymptotic formula for Stancu-type generalization of Jain-Beta operators
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