On the best approximation properties of $C^\infty$-smooth functions on an interval of the real axis (to the phenomenon of unsaturated numerical methods)

In 1975 K. I. Babenko announced his discovery of conceptually new unsaturated numerical methods. They are distinguished by the absence of the principal error term, which results in their ability to adjust automatically to all natural correctness classes of problems (the phenomenon of unsaturated numerical methods).
We show that the phenomenon of unsaturation of a numerical method on an interval is a consequence, although exceptionally subtle, of the well-developed theory of polynomial approximation to continuous functions. By the way, K. I. Babenko always insisted on that.