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On simultaneous approximation by modified Lupas operators. (English) Zbl 0596.41035

A ”modified Lupas operator” is defined, for functions integrable in [0,+), in the following way:

(M n f)(x)=(n-1) k=0 P n,k (x) 0 P n,k (y)f(y)dy,

where

P n,k (t)=n+k-1 kt k /(1+t) n+k ·

Two general theorems, concerning the approximation of functions with these operators, are given. If we set A r =M n (r) f(x)-f (r) (x), then we can say that the first theorem deals with lim n |A r |, when f has a derivative of (r+2)th order; the second one gives an estimate of A r in terms of the modulus of continuity of f (r) .

Reviewer: P.L.Papini

MSC:
41A36Approximation by positive operators
41A25Rate of convergence, degree of approximation