Affiliation:
1. Department of Mathematics, Aligarh Muslim University, Aligarh, India
2. Department of Mathematics, Aligarh Muslim University, Aligarh, India + Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan
Abstract
In this paper, Lupa? Bernstein-Kantorovich operators have been studied using
Jackson and Riemann type (p,q)-integrals. It has been shown that (p,
q)-integrals as well as Riemann type (p, q)-integrals are not well defined
for 0 < q < p < 1 and thus further analysis is needed. Throughout the paper,
the case 1 ? q < p < ? has been used. Advantages of using Riemann type (p,
q)-integrals are discussed over general (p, q)-integrals. Lupa?
Bernstein-Kantorovich operators constructed via Jackson integral need not be
positive for every f ? 0. So to make these operators based on general (p,
q)-integral positive, one need to consider strictly monotonically increasing
functions, and to handle this situation Lupa? Bernstein-Kantorovich
operators are constructed using Riemann type (p, q)-integrals. However
Lupa? (p, q)-Bernstein-Kantorovich operators based on Riemann type (p,
q)-integrals are always positive linear operators. Approximation properties
for these operators based on Korovkin?s type approximation theorem are
investigated. The rate of convergence via modulus of continuity and function
f belonging to the Lipschitz class is computed.
Publisher
National Library of Serbia
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