On Smoothness and Approximation Properties of the Kantorovich Type Operators

Abstract

Abstract. In this paper we present inequalities concerning the weighted moduli of continuity of Kantorovich type operators L^f-Moreover, we give some estimates of the degree of approximation of / by Lnf in the Holder type norms. PreliminariesLet I be a finite or infinite interval and let M(I) be the class of all measurable complex-valued functions bounded on I. In the case when I is an infinite interval, denote by M\ oc (I) the class of all functions bounded on every compact subinterval of I. Given any n G N :-{1,2,...}, let J n be a set of indices contained in Z := {0, ±1, ±2,...} and let I be the union of non-overlapping intervals Ij !n (j £ Jn) with increasing left (right) end points. Introduce, formally, for functions / belonging to M{I) or M\ oc {I), the discrete operators L n defined byjeJ n where € ij, n and pj >n are non-negative functions continuous on I. Denote by L* n the Kantorovich type modification of operators (1.1) given by We adopt the following notation. Given any non-negative function w defined on the interval ICR and any x,y £ I we writeFor an arbitrary function g defined on I we introduce the quantityWe denote by A (I) the set of all continuous functions w on I, positive in the interior of I, with values not greater than 1, which satisfy the inequality w(x,y) < w(s) for any three points x, s,y G I such that x < s < y.(Obviously, this inequality holds if, for example, w is non-decreasing or nonincreasing or concave on I). Given two weights w,p G A(J) we define the general weighted modulus of continuity of g on I byIf p(x) = 1 on /, we will write fi w (g; 6) instead of £l w ,p(9', Further, in the case w(x) = 1 on I the weighted modulus il w (g~,6) becomes the ordinary modulus of continuity u(g\ 6) of g on I. In this case, w = 1, the symbol ||(?|| will be used instead of HffH^. Let C W (I) denote the class of all functions g continuous on I, such that ||<7||u, < 00.Taking into account a positive non-decreasing function tp on the interval (0,1], such that ip( 1) < 1, we write iuil(v) . luii• --f 19{x) ~ 9(y)\w{x,y)p{x,y)x,y G I, 0 < \x-y\ < lj.If this quantity is finite we call it the weighted Holder type norm of g on I.In case p = 1 and w = 1 on /, we denote this expression by if p # 1, w = 1, then we will write ||<7||p^ instead of ||<?||&v>.