Abstract
The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $ -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q ) $ -integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed.
Highlights
The (p, q)-calculus is a generalization of the well-known q-calculus and it is constructed by the following notations and definitions
In Section, we investigate the uniform convergence of the operators and present the rate of convergence via the weighted modulus of continuities
Proof According to the weighted Korovkin theorem proved in [ ], it is sufficient to verify the following three conditions: lim n→∞
Summary
The (p, q)-calculus is a generalization of the well-known q-calculus and it is constructed by the following notations and definitions. Let f : C[ , a] → R (a > ) the (p, q)-integration of a function f is defined by a. To approximate the functions via polynomials based on (p, q)-integers, no doubt, would have a crucial role To fulfill this necessity, very recently the well-known sequences of linear positive operators of approximation theory have been transferred to the (p, q)-calculus and the advantages of (p, q) analogs of them have been intensively investigated. Aral and Gupta [ ] introduced the (p, q)-analog of the well-known Baskakov operators by Another problem in the approximation by linear positive operators is to present an approximation process for Riemann integrable functions. Definition For x ∈ [ , ∞), < q < p ≤ , the (p, q)-analog of the Baskakov-Kantorovich operators is defined as. B ∗n,p,q(g; x) – g(x) ≤ g CB α∗(n)( + x) + βn (p, q, x)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
