Approximation Properties of Generalized λ -Bernstein–Stancu-Type Operators

Abstract

The present study introduces generalized <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:mi>λ</a:mi> </a:math> -Bernstein–Stancu-type operators with shifted knots. A Korovkin-type approximation theorem is given, and the rate of convergence of these types of operators is obtained for Lipschitz-type functions. Then, a Voronovskaja-type theorem was given for the asymptotic behavior for these operators. Finally, numerical examples and their graphs were given to demonstrate the convergence of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:msubsup> <c:mrow> <c:mi>G</c:mi> </c:mrow> <c:mrow> <c:mi>m</c:mi> <c:mo>,</c:mo> <c:mi>λ</c:mi> </c:mrow> <c:mrow> <c:mi>α</c:mi> <c:mo>,</c:mo> <c:mi>β</c:mi> </c:mrow> </c:msubsup> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>f</c:mi> <c:mo>,</c:mo> <c:mi>x</c:mi> </c:mrow> </c:mfenced> </c:math> to <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M4"> <h:mi>f</h:mi> <h:mfenced open="(" close=")" separators="|"> <h:mrow> <h:mi>x</h:mi> </h:mrow> </h:mfenced> </h:math> with respect to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M5"> <m:mi>m</m:mi> </m:math> values.