A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α
M. Mursaleen, Nadeem Rao, Mamta Rani, Adem Kılıçman, Ahmed Ahmed Hussin Ali Al-Abied, Pradeep Malik
Abstract
The objective of this paper is to construct univariate and bivariate blending type <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>α</a:mi></a:math> -Schurer–Kantorovich operators depending on two parameters <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>α</c:mi><c:mo>∈</c:mo><c:mfenced open="[" close="]" separators="|"><c:mrow><c:mn>0,1</c:mn></c:mrow></c:mfenced></c:math> and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"><h:mi>ρ</h:mi><h:mo>></h:mo><h:mn>0</h:mn></h:math> to approximate a class of measurable functions on <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"><j:mfenced open="[" close="]" separators="|"><j:mrow><j:mn>0,1</j:mn><j:mo>+</j:mo><j:mi>q</j:mi></j:mrow></j:mfenced><j:mo>,</j:mo><j:mi>q</j:mi><j:mo>></j:mo><j:mn>0</j:mn></j:math> . We present some auxiliary results and obtain the rate of convergence of these operators. Next, we study the global and local approximation properties in terms of first- and second-order modulus of smoothness, weight functions, and by Peetre’s <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M5"><o:mi>K</o:mi></o:math> -functional in different function spaces. Moreover, we present some study on numerical and graphical analysis for our operators.