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Positive linear operators which preserve x 2 . (English) Zbl 1027.41028
The approximations of continuous functions f on [0,1] by a sequence of positive linear operators L n always converge to f iff L n preserve the three functions e i (x)=x, i=0,1,2 (Korovkin theorem). Replacing the variable x in the Bernstein polynomials by some functions r n (x) the author defines the operators L n acting on 𝒞([0,1]), satisfying the Korovkin condition and leading to the order of approximation of f at least as good as the order of approximation by Bernstein polynomials. The summability matrix A is defined by means of the functions r n (x) and it is proved that A preserves the limits of complex sequences provided lim n r n (x)=x.

MSC:
41A40Saturation (approximations and expansions)
40G99Special methods of summability