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 {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 > α > k 0 > \alpha > k between E n ( f ) p = O ( n − α ) {E_n}{(f)_p} = O({n^{ - \alpha }}) and ω φ k ( f , t ) p = O ( t α ) \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 ⩽ p ⩽ ∞ {L_p}[ - 1,1],\;1 \leqslant p \leqslant \infty . Analogous results for approximating periodic functions by trigonometric polynomials in L p [ − π , π ] , 0 > p ⩽ ∞ {L_p}[ - \pi ,\pi ],0 > p \leqslant \infty , are known.