Abstract
In the present paper, the generalized p,q-gamma-type operators based on p,q-calculus are introduced. The moments and central moments are obtained, and some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K-functional. Also, the rate of convergence, weighted approximation, and pointwise estimates of these operators are studied. Finally, a Voronovskaja-type theorem is presented.
Highlights
In [1], Mazhar introduced gamma operators preserving linear functions as follows: ð∞Gnð f ; xÞ = gnðx ; uÞdu gn−1ðu ; tÞf ðtÞdt = ð2nÞ!xn−1 ð∞n!ðn − 1Þ! 0 tn−1 ðx + tÞ2n+1 f ðtÞdt, ð1Þ where gnðx ; uÞ = ðxn+1/n!Þe−xuun, n > 1, x > 0
The paper is organized as follows: In Section 1, we introduce the history of gamma-type operators and recall some basic notations about ðp, qÞ-calculus; we construct the generalized ðp, qÞ-gamma operators with the ðp, qÞ-beta function
We prove the limit equality (19) while k ≥ 4, k = 1, 2, 3 is similar
Summary
In [1], Mazhar introduced gamma operators preserving linear functions as follows: ð∞. In [3], Mao defined generalized gamma operators as follows: Gn,kð f ; xÞ = gnðx ; uÞdu gn−kðu ; tÞf ðtÞdt ð2n. In [35], Cheng and Zhang constructed a ðp, qÞ-analogue of the operators (1) using the ðp, qÞ-beta function of the second kind and studied their approximation properties. Cheng et al defined the ðp, qÞ-analogue of the operators (2) and researched their approximation properties in [36] All these achievements motivate us to construct the ðp, qÞ-analogue of the gamma operator (3) and generalize the works of [35, 36]. The paper is organized as follows: In Section 1, we introduce the history of gamma-type operators and recall some basic notations about ðp, qÞ-calculus; we construct the generalized ðp, qÞ-gamma operators with the ðp, qÞ-beta function.
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