Universal properties of approximation operators

Abstract

We discuss universal properties of some operators L n : C [ 0 , 1 ] → C [ 0 , 1 ] . The operators considered are closely related to a theorem of Korovkin (1960) [4] which states that a sequence of positive linear operators L n on C [ 0 , 1 ] is an approximation process if L n f i → f i ( n → ∞ ) uniformly for i = 0 , 1 , 2 , where f i ( x ) = x i . We show that L n f may diverge in a maximal way if any requirement concerning L n in this theorem is removed. There exists for example a continuous function f such that ( L n f ) n ∈ N is dense in ( C [ 0 , 1 ] , ‖ . ‖ ∞ ) , even if L n is positive, linear and satisfies L n P → P ( n → ∞ ) for all polynomials P with P ( 0 ) = 0 .