65 years since the paper "on the value of the best approximation of functions having a real singular point" by I. I. Ibragimov

In his famous paper [15] Ibragim Ibishievich Ibragimov has given asymptotic values of the best uniform approximation of functions of the form (a - x)^s ln^m(a - x), (a ≥ 1). These results have led to the development of a series of new directions in approximation theory, including the following ones, to which we devote this paper. • Constructive characterization of approximation of functions on a closed interval. • Babenko spaces. • Ditzian-Totik moduli of smoothness. • Constructive characterization of approximation of functions on the sets of complex plane. • Shape preserving approximation. In particular, we will show how we have used the results by I. I. Ibragimov in our recent paper in Constructive Approximation.