A complete asymptotic expansion for operators of exponential type with pleft( xright) =xleft( 1+xright) ^{2}

Ulrich Abel,Vijay Gupta

doi:10.1007/s11117-020-00802-5

Ulrich Abel, Vijay Gupta

Open Access

https://doi.org/10.1007/s11117-020-00802-5

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Abstract

In the year 1978, Ismail and May studied operators of exponential type and proposed some new operators which are connected with a certain characteristic function pleft( xright) . Several of these operators were not separately studied by researchers due to its unusual behavior. The topic of the present paper is the local rate of approximation of a sequence of exponential type operators R_{n} belonging to pleft( xright) =xleft( 1+xright) ^{2}. As the main result we derive a complete asymptotic expansion for the sequence left( R_{n}fright) left( xright) as n tends to infinity.

Highlights

Let Cγ [0, ∞) be the class of continuous functions f on [0, ∞) satisfying the growth condition f (t) = O eγ t as t → ∞, for some γ > 0.U
In 1972 Jain [12] introduced the sequence of linear operators defined, for functions f ∈ C [0, +∞), by
The main result is a complete asymptotic expansion for the sequence of the Ismail–May operators (Rn f ) (x) as n tends to infinity

Summary

Introduction

In 1972 Jain [12] introduced the sequence of linear operators defined, for functions f ∈ C [0, +∞), by. → on 0 as n → ∞ the each compact subset of [0, +∞) His proof is based on Korovkin’s theorem and does not consider any condition on the growth rate of the function f as x → +∞. Both operators (1) and (4) are principally different, because in the second definition β tends to zero as n → ∞. We study the local rate of convergence of the operators Rn as n tends to infinity. We study the asymptotic behaviour of the sequence ((Rn f ) (x))∞ n=1 as n tends to infinity. We apply concise expressions for the moments derived in [1] by using methods of complex analysis

Main results

Auxiliary results