ON RESTRICTED BEST APPROXIMATION TO FUNCTIONS WITH RESTRICTED DERIVATIVES.

Let a function f belonging to C**a left bracket minus 1,1 right bracket (where a equals 2k//r, for some fixed k//r greater than equivalent to 0), be such that the sum over n of (1/n) omega (f**a, 1/n** one-half ) less than infinity . It is shown that if f satisfies r restrictions on its 0 less than equivalent to k//1 less than k//2 less than . . . less than k, derivatives respectively with strict inequalities, then for sufficiently large n, the best polynomial approximator to f satisfies the same restriction. Thus the best polynomial approximator is also the best restricted derivatives approximator.