Abstract
In the present paper, we define a generalization sequence of linear positive operators based on four parameters which is reduce to many other sequences of summation–integral older type operators of any weight function (Bernstein, Baskakov, Szász or Beta). Firstly, we find a recurrence relation of the -th order moment and study the convergence theorem for this generalization sequence. Secondly, we give a Voronovaskaja-type asymptotic formula for simultaneous approximation. Finally, we introduce some numerical examples to view the effect of the four parameters of this sequence.
Highlights
In the past three years, many researchers have been built and studied sequences of summation-integral type operators based on parameters
We introduce a Voronovskaja type asymptotic formula for the sequence Hαρ(f; x, c, r), we show that the s-th derivative ds dws
The results are explain by graphs and the error functions occur between the test functions and the approximations
Summary
In the past three years, many researchers have been built and studied sequences of summation-integral type operators based on parameters. These sequences give us some older sequence of summation-integral type operators, when we give suitable values for the parameters This paper is a continuation of the work of previous papers [1, 2, 3, 4, 5, 6, 8 and 11]. We define and study our sequence based on four parameters ρ > 0, c ∈ N0 ∪ {−1} and r ∈ N0 = N ∪ {0} as follows: Hαρ(f, x, c, r) = ∑ pα,k(x, c, r) ∫ θαρ,k(t)f(t)dt , x ∈ [0, ∞), k=0 where pα,k(x, c, r). We assume that M is a positive real constant not necessarily the same in different cases
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