Approximation by operators of probabilistic type

Abstract This paper is concerned with sets W of sequences ( W ) n = 1 ∞ ϵ W of positive linear operators which are of certain probabilistic type and act on certain function classes K . Necessary and sufficient conditions upon W are determined such that each element ( W n ) n = 1 ∞ ϵ W approximates U with a given order of approximation ψ ( n ) and a given function class K , the limiting operator U being either the identity I or an operator connected with the normal distribution. The saturation problem in this setting is also solved, now in a form giving the order of saturation ψ ( n ) such that convergence of ( W n ) n = 1 ∞ towards U of order o ( ψ ( n )), n → ∞ impossible unless W n = I , n ϵ N , and there exists a non-trivial element ( W n ) n = 1 ∞ which approximates U with order O ( ψ ( n )).