Source-linked AI summary
Some Approximation Results by $(p,q)$-analogue of Bernstein-Stancu Operators (Revised)
M. Mursaleen, Khursheed J. Ansari, Asif Khan
TL;DR
The paper addresses a defect in an earlier Bernstein-Stancu construction, where the operators fail to reproduce the constant function across most of [0,1]. It introduces revised (p,q)-Bernstein-Stancu operators and proves, via the Bohman-Korovkin theorem, uniform convergence to f on [0,1] under specified sequences pn and qn.
Problem
The earlier operators fail to satisfy S_n,p,q(1;x)=1 for x∈[0,1), motivating a revised construction.
Method
The paper defines revised (p,q)-Bernstein-Stancu operators using (p,q)-integers and establishes approximation properties through the Bohman-Korovkin theorem.
Results
S_n,p,q(f,x) converges uniformly to f on [0,1] when the sequences pn and qn satisfy the stated conditions.
Takeaways & Limitations
The revised operators provide a uniform approximation framework that includes (p,q)-Bernstein and q-Bernstein-Stancu operators as special cases.
Abstract
from arXiv · showhide
In this paper, we have given a corrigendum to our paper "Some Approximation Results by $(p,q)$-analogue of Bernstein-Stancu Operators" published in Applied Mathematics and Computation $264 (2015) 392-402.$ We introduce a new analogue of Bernstein-Stancu operators and we call it as $(p,q)$-Bernstein-Stancu operators. We study approximation properties based on Korovkin's type approximation theorem of $(p,q)$-Bernstein-Stancu operators. We also establish some direct theorems.
1 Construction of Revised Operators
The paper reintroduces the Bernstein-Stancu operators after identifying that the earlier form fails to reproduce the constant function on most of [0,1]. The revised construction is defined using (p,q)-integers and includes established Bernstein variants as special cases.
- Revised operators: The revised (p,q)-Bernstein-Stancu operators are introduced as a new analogue of Bernstein-Stancu operators.
- Motivation for revision: The earlier operators do not satisfy S_n,p,q(1; x)=1 for x∈[0,1), motivating their reintroduction.They reproduce 1 only at x=1.
- Special cases: For α=β=0, the revised operators reduce to (p,q)-Bernstein operators, while p=1 yields q-Bernstein-Stancu operators.
- Notation: The construction uses (p,q)-integers [n]p,q, with q-integers recovered when p=1 and ordinary integers when p=q=1.
- Basic properties: A basic lemma is established using the operator's linearity and the recurrence [k+1]p,q=pk+q[k]p,q.
2 Main Results:
The main approximation result studies sequences pn and qn satisfying the stated parameter conditions and applies the Bohman-Korovkin theorem to the test functions 1, x, and x^2. Under these conditions, the operators converge uniformly to the target function on [0,1].
- Parameter conditions: For q∈(0,1) and p∈(q,1], the parameter sequences are chosen to satisfy the limiting conditions stated in Remark 2.1.
- Uniform approximation: Theorem 3.1.1 states that S_n,p,q(f,x) converges uniformly to f on [0,1] for sequences satisfying the required remarks.
- Korovkin argument: The convergence proof applies the Bohman-Korovkin theorem by checking the operators on t^m for m=0,1,2.
- Proof estimates: The proof uses bounds derived from Lemma 2.1 and takes their maximum to obtain the required convergence estimates.
- Proof completion: The argument concludes after establishing the stated test-function limits, with remaining results asserted to follow similarly.