A new characterization of weighted Peetre \(K\)-functionals (Q1772223)

B. R. Draganov and K. G. Ivanov present several results related with the problem of finding moduli of smoothness equivalent to certain weighted Peetre K-functionals for functions of a real variable. The main idea is to characterize the original K-functional by another one obtained with the help of an appropriated operator. The method works well when the new K-functional is equivalent to a known modulus of smoothness. Thus the authors study in details operators which change the weight in the K-functionals and preserve the smoothness properties of the functions. The new approach has several advantages. For instance it is not necessary to consider the so called main-part of the modulus, as in the case of Ditzian-Totik type moduli. In the last sections of the paper the authors give concrete examples of operators and application to the problem of best algebraic polynomial approximation and approximation by linear operators. The paper is well written and I think it will be very useful for people in approximation theory.