The famous Korovkin Theorem states, that if the sequence of Positive Linear Operators Ln uniformly approximates the test functions 1, t, t2 , then it follows that Ln uniformly approximates each continuous function f(x), defined on closed bounded interval. Is it true, that the number 3 of test functions, can be replaced by 2 or 1 test function (may be different from 1, t, t2 ) such that we still have uniform convergence? If yes-please give example, if no-please give conterexample, or relevant references?
No, 3 is the smallest cardinality of the test set. This was shown by Korovkin himself, see his 1959 "Linear operators and approximation theory" monograph, Section 5, Corollary 2 (p.47 of Russian version or p.41 of English translation). Thanks go to Farzaneh Jannat for providing the answer.
Dear prof.Prymak, many thanks for the letter! Also thanks are going to Farzaneh Jannat! Your fast reply closes now the discussion of my question. With best wishes! G.Tachev