Approximation with an arbitrary order by generalized Kantorovich-type and Durrmeyer-type operators on [0;+\infinit)

Abstract

Given an arbitrary sequence ¸n > 0, n 2 N, with the prop- erty that limn!1 ¸n = 0 as fast we want, in this note we introduce modi¯ed/generalized Sz¶asz-Kantorovich, Baskakov- Kantorovich, Sz¶asz- Durrmeyer-Stancu and Baskakov-Sz¶asz-Durrmeyer-Stancu operators in such a way that on each compact subinterval in [0;+1) the order of uniform approximation is !1(f; p ¸n). These modi¯ed operators uni- formly approximate a Lipschitz 1 function, on each compact subinterval of [0;1) with the arbitrary good order of approximation p ¸n. The re- sults obtained are of a de¯nitive character (that is are the best possible) and also have a strong unifying character, in the sense that for vari- ous choices of the nodes ¸n, one can recapture previous approximation results obtained for these operators by other authors.