Generalized Kantorovich operators

Abstract In this paper we will propose a class of generalized Kantorovich type operators constructed using a general differential operator with non-constant coefficients <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mi>l</m:mi> </m:msup> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mi>l</m:mi> </m:msubsup> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mi>i</m:mi> </m:msub> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>i</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {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.