[Paper Review] Approximation properties of (p;q)-variant of Stancu-Schurer operators(Revised)

Abdul Wafi, Nadeem Rao|arXiv (Cornell University)|Aug 8, 2015

Approximation Theory and Sequence Spaces8 references9 citations

TL;DR

This paper introduces a new (p,q)-variant of Stancu-Schurer operators that generalizes classical and q-analogue operators in approximation theory. It establishes convergence rates using second-order modulus of continuity, Peetre's K-functional, and Lipschitz classes, proving direct approximation theorems with explicit error bounds in terms of (p,q)-integers and parameters α, β.

ABSTRACT

In this article, we have introduced (p;q)-variant of Stancu-Schurer operators and discussed the rate of convergence for continuous functions. We have also discussed recursive estimates Korovkin and direct approximation results using second order modulus of continuity, Peetres K-functional Lipschitz class.

Motivation & Objective

• To extend Bernstein-Schurer and Stancu operators to a (p,q)-calculus framework for improved approximation flexibility.
• To analyze the rate of convergence of the new (p,q)-variant operators for continuous functions on [0,1].
• To establish direct approximation results using second-order modulus of continuity, Peetre’s K-functional, and Lipschitz classes.
• To generalize existing results in q-approximation theory by incorporating two parameters p and q.

Proposed method

• Define the (p,q)-variant of Stancu-Schurer operators using (p,q)-integers and (p,q)-binomial coefficients.
• Derive moment estimates for the operators using recursive identities involving [n]_p,q and p,q-binomial expansions.
• Apply Korovkin’s theorem to prove convergence for continuous functions on [0,1].
• Use second-order modulus of continuity and Peetre’s K-functional to establish direct approximation estimates.
• Employ Hölder’s inequality and Lipschitz class conditions to bound the approximation error for functions in Lip_M(α).
• Prove convergence rate bounds in terms of δ_n,l^α,β(x) = S_n,l^α,β((t−x)^2; x) for functions in Lip_M(α).

Experimental results

Research questions

• RQ1 How do the (p,q)-variant Stancu-Schurer operators converge for continuous functions on [0,1]?
• RQ2 What is the rate of convergence of these operators using second-order modulus of continuity and Peetre’s K-functional?
• RQ3 How do the parameters α and β affect the approximation error in the (p,q)-framework?
• RQ4 Can the Lipschitz class condition be used to derive explicit error bounds for the (p,q)-operators?
• RQ5 What is the relationship between the new (p,q)-operators and existing q-analogue and classical Bernstein-Schurer operators?

Key findings

• The (p,q)-variant Stancu-Schurer operators converge uniformly to continuous functions on [0,1] as n→∞, satisfying Korovkin’s theorem.
• The rate of convergence is bounded by a constant multiple of the second-order modulus of continuity: |S_n,l^α,β(f;x)−f(x)| ≤ C ω_2(f; √Θ_n,l^α,β(x)).
• For functions in the Lipschitz class Lip_M(α), the error is bounded by M (δ_n,l^α,β(x))^{α/2}, where δ_n,l^α,β(x) = S_n,l^α,β((t−x)^2; x).
• The approximation error also depends on the parameter ratio [n+l]_p,q / ([n]_p,q + β), which tends to 1 as n→∞, ensuring convergence.
• The results generalize Mursaleen et al.’s (p,q)-Bernstein-Schurer operators, reducing to their case when α=β=0.
• The use of Peetre’s K-functional leads to a refined error estimate combining smoothness and operator moments.