[Paper Review] Some approximation results by Bernstein-Kantorovich operators based on (p,q)-integers

M. Mursaleen, Khursheed J‎. ‎Ansari+1 more|arXiv (Cornell University)|Apr 16, 2015

Approximation Theory and Sequence Spaces17 references5 citations

TL;DR

This paper introduces $(p,q)$-Bernstein-Kantorovich operators as a generalization of $q$-analogue operators, enhancing flexibility in function approximation. Using Korovkin's theorem and modulus of continuity, the authors establish convergence rates and local approximation properties, demonstrating superior convergence flexibility over $q$-operators through theoretical analysis and graphical comparisons.

ABSTRACT

In this paper, First we have given the modified form of (p,q)-analogues of Bernstein and Bernstein operators [21-23] and then we introduce a new analogue of Bernstein-Kantorovich operators which we call as (p,q)-Bernstein-Kantorovich operators. We discuss approximation properties for these operators based on Korovkin's type approximation theorem and we compute the order of convergence using usual modulus of continuity and also the rate of convergence when f is a Lipschitz function. Moreover, we also study the local approximation property of the (p,q)-Kantorovich operators . We show comparisons and some illustrative graphics for the convergence of operators to a function. In comparison to q-analogoue of Bernstein-Kantorovich operators, our generalization gives more flexibility for the convergence of operators to a function.

Motivation & Objective

• To develop a new generalization of Bernstein-Kantorovich operators using $(p,q)$-calculus for improved approximation flexibility.
• To analyze the approximation properties of the proposed $(p,q)$-Bernstein-Kantorovich operators using Korovkin-type theorems.
• To compute the order of convergence via modulus of continuity and rate of convergence for functions in the Lipschitz class $\mathrm{Lip}_M(\alpha)$.
• To investigate the local approximation behavior of the sequence $K_n^{(p,q)}$ using second-order $K$-functional and Peetre's $K$-functional.

Proposed method

• Propose a new operator $K_n^{(p,q)}(f;x)$ based on $(p,q)$-integers, generalizing $q$-Bernstein-Kantorovich operators.
• Define the $(p,q)$-calculus framework, including $(p,q)$-integers, $(p,q)$-binomial coefficients, and $(p,q)$-integrals.
• Establish a basic lemma for moment estimates of the operator $K_n^{(p,q)}$ to support convergence analysis.
• Apply Korovkin’s theorem to prove uniform convergence on $[0,1]$ for continuous functions.
• Use the modulus of continuity and $K$-functional to derive quantitative estimates for the rate of convergence.
• Generate graphical comparisons using MATLAB to illustrate convergence behavior for different $p$, $q$, and $n$ values.

Experimental results

Research questions

• RQ1 How do $(p,q)$-Bernstein-Kantorovich operators improve convergence flexibility compared to $q$-analogs in function approximation?
• RQ2 What is the order of convergence of the proposed $(p,q)$-operators using the usual modulus of continuity?
• RQ3 What is the rate of convergence when the function $f$ belongs to the Lipschitz class $\mathrm{Lip}_M(\alpha)$?
• RQ4 How does the local approximation property of $K_n^{(p,q)}$ behave, and what is its quantitative estimate?
• RQ5 To what extent do graphical examples demonstrate the superiority of $(p,q)$-operators over $q$-operators in convergence speed and accuracy?

Key findings

• The proposed $(p,q)$-Bernstein-Kantorovich operators converge uniformly to continuous functions on $[0,1]$ as $n \to \infty$, satisfying Korovkin’s conditions.
• The order of convergence is estimated as $O(\delta_n^2(x))$ using the modulus of continuity, where $\delta_n^2(x)$ is the second moment of the operator.
• For functions in $\mathrm{Lip}_M(\alpha)$, the rate of convergence is $O\left(\left(\frac{p^n}{[n]_{p,q}}\right)^{\alpha/2}\right)$, showing dependence on $p$, $q$, and $n$.
• The local approximation error is bounded by $C \omega_2(f, \delta_n(x)) + \omega(f, \alpha_n(x))$, where $\omega_2$ is the second-order modulus of continuity and $\alpha_n(x) = \frac{p^n}{[2]_{p,q}[n]_{p,q}}$.
• Graphical results confirm that as $p \to 1^-$ and $q \to 1^-$ with $0 < q < p \leq 1$, the operators converge more flexibly and rapidly to $f(x) = 1 + \sin(7x)$ than $q$-analogs.
• The $K$-functional approach yields a quantitative estimate: $\left|K_n^{(p,q)}(f;x) - f(x)\right| \leq 4K_2(f, \delta_n^2(x)) + \omega\left(f, \frac{p^n}{[2]_{p,q}[n]_{p,q}}\right)$, confirming convergence with optimal smoothness dependence.