ON RESTRICTED BEST APPROXIMATION TO FUNCTIONS WITH RESTRICTED DERIVATIVES

Let a function $f \in C^{(2k_r )} [ - 1,1]$, for some fixed $k_r \geqq 0$, be such that $\sum _n {({1 / n})} \omega (f^{(2k_r )} ,{1 / {\sqrt n }}) < \infty $. We show that if f satisfies r restrictions on its $0 \leqq k_1 < k_2 < \cdots < k_r $ 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.