Analog of Bernstein's theorem in the space \(L_ 1\) (Q1909785)

The S. N. Bernstein's theorem giving the necessary and sufficient condition guaranteed that a continuous function \(f\) has a continuous derivative of order \(k\) in some interval \(l\) is translated in the ``global'' analogue by means of a subdivision of the interval \(l\) completed by some subsequent convergence conditions. This trick allows to enlarge a class of functions under consideration which can be characterised by a Bernstein-like theorem.