Equivalence Theorem on Weighted Simultaneous Lp-Approximation by the Method of Kantorovič Operators

AbstractThe aim of the present paper is to prove new direct and converse results on weighted simultaneous approximation by the method of classical Kantorovič operators in the Lp-metric using the weighted Ditzian and Totik modulus of smoothness. We will obtain with a direct technique for integrable functions the complete equivalence characterization of weighted simultaneous approximation by this method (Kn)n∈ N. One of the main tools and crucial estimates managing the converse and equivalence results of the simultaneous behaviour is given by a direct modified Voronovskaja theorem which uses the third order weighted modulus of smoothness