On the order of an approximation of functions on sets of positive measure by linear positive polynomial operators

It is proved that at almost all points the order of approximation, even of one of the functions 1, cos x,sin x by means of a sequence of linear positive polynomial operators having uniformly bounded norms, is not higher than 1/n². Refinements of this result are given for operators of convolution type.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 457–468, March, 1973.In conclusion the author expresses thanks to P. P. Korovkin for posing the problem.