Inverse theorem for best polynomial approximation in $L_ p,\;0<p<1$

Abstract:

A direct theorem for best polynomial approximation of a function in ${L_p}[ - 1,1],\;0 < p < 1$, has recently been established. Here we present a matching inverse theorem. In particular, we obtain as a corollary the equivalence for $0 < \alpha < k$ between ${E_n}{(f)_p} = O({n^{ - \alpha }})$ and $\omega _\varphi ^k{(f,t)_p} = O({t^\alpha })$. The present result complements the known direct and inverse theorem for best polynomial approximation in ${L_p}[ - 1,1],\;1 \leqslant p \leqslant \infty$. Analogous results for approximating periodic functions by trigonometric polynomials in ${L_p}[ - \pi ,\pi ],0 < p \leqslant \infty$, are known.