Generalized Kantorovich operators

In this paper we will propose a class of generalized Kantorovich type operators constructed using a general differential operator with non-constant coefficients Dlg(x)=∑i=0lai(x)g(i)(x) {D^l}g\left( x \right) = \sum\nolimits_{i = 0}^l {{a_i}} \left( x \right){g^{\left( i \right)}}\left( x \right) and its corresponding antiderivative operator I^l with respect to the composition D^l ◦ I^l = Id. The operators studied are of the type K_n = D^l ◦ L_n ◦ I^l where L_n are positive linear operators. For these operators we will prove an approximation result and a Voronovskaja type theorem. Also, a simultaneous approximation result will be provided for a particular case. The operators studied in this paper are linear but not positive operators.