ABSTRACT
The first publication of S. N. Bernstein dates back to 1903. He proved Hilbert’s theorem in a form more general than its original version; this was possible because the special structure of equation also happens to be unimportant. S. N. Bernstein writes that his proof was obtained by extending and modifying Picard’s method. It is also possible to examine, from a general viewpoint, the difference between the two criteria of analyticity: one in terms of the Picard norm being finite, and another in terms of the Bernstein norm being finite. There is a large group of S. N. Bernstein’s works dedicated to the first boundary value problem (the Dirichlet problem) for nonlinear equations. A very simple proof of Efimov’s theorem was found by E. Heinz, who used an identity obtained by S. N. Bernstein. We briefly consider the technique developed by S. N. Bernstein for the derivation of a priori estimates.
