Approximation with an arbitrary order by modified Baskakov type operators

Abstract

Given an arbitrary sequence λn > 0, n∈N, with the property that limn→∞λn=0 so fast as we want, in this note we consider several kinds of modified Baskakov operators in which the usual knots jn are replaced with the knots j · λn. In this way, on each compact subinterval in [0,+∞) the order of uniform approximation becomes ω1(f;λn). For example, these modified operators can uniformly approximate a Lipschitz 1 function, on each compact subinterval of [0, ∞) with the arbitrary good order of approximation λn. Also, similar considerations are made for modified qn-Baskakov operators, with 0 < qn < 1, limn→∞qn=1.