(?, ?)-Bernstein-Kantorovich operators

Abstract

In this article, we introduce a new family of ( lambda , psi ) \left(\lambda ,\psi ) -Bernstein-Kantorovich operators which depends on a parameter lambda \lambda , derived from the basis functions of B & eacute;zier curves and an integrable function psi \psi . In this approach, all moments and central moments of the new operators can be obtained in terms of two numbers M 1 , psi {M}_{1,\psi } and M 2 , psi {M}_{2,\psi } , which are the integrals of psi \psi and psi 2 {\psi }<^>{2} , respectively. For operators L n ( f ; x ) {L}_{n}\left(f;\hspace{0.33em}x) with L n ( 1 ; x ) = 1 {L}_{n}\left(1;\hspace{0.33em}x)=1 , the order of approximation to a function f f by L n ( f ; x ) {L}_{n}\left(f;\hspace{0.33em}x) is more controlled by the term L n ( ( t - x ) 2 ; x ) {L}_{n}\left({\left(t-x)}<^>{2};\hspace{0.33em}x) . For our operators K n , lambda , psi ( f ; x ) {K}_{n,\lambda ,\psi }\left(f;\hspace{0.33em}x) , the second central moment K n , lambda , psi ( ( t - x ) 2 ; x ) {K}_{n,\lambda ,\psi }\left({\left(t-x)}<^>{2};\hspace{0.33em}x) depend on M 1 , psi {M}_{1,\psi } and M 2 , psi {M}_{2,\psi } . This means that in our new approach, it is possible to search for a function psi \psi with different values of M 1 , psi {M}_{1,\psi } and M 2 , psi {M}_{2,\psi } to make K n , lambda , psi ( ( t - x ) 2 ; x ) {K}_{n,\lambda ,\psi }\left({\left(t-x)}<^>{2};\hspace{0.33em}x) smaller. Using this new approach, we show that there exists a function psi \psi such that the order of approximation to a function f f by our new ( lambda , psi ) \left(\lambda ,\psi ) -Bernstein-Kantorovich operators is better than the classical lambda \lambda -Bernstein-Kantorovich operators on the interval [0, 1]. Moreover, we obtain some direct and local approximation properties of new operators. We also show that our operators preserve monotonicity properties. Furthermore, we illustrate the approximation results of our operators graphically and numerically.