Polynomial approximation with doubling weights

Abstract

Among other things, we prove that, for a doubling weight $w$, $0<p\le \infty$, $r\in {\mathbb{N}}_0$, and $0<\alpha <r+1-1/{\lambda }_p$, we have \[ E_n(f)_{p, w_n} = O(n^{-α}) \iff ω_φ^{r+1}(f, n^{-1})_{p, w_n} = O(n^{-α}), \] where ${\lambda }_p:=p$ if $0<p<\infty$, ${\lambda }_p:=1$ if $p=\infty$, $\Vert f{\Vert }_{p,w}:={\left({\int }_{-1}^1|f(u){|}^{p}w(u)du\right)}^{1/p}$, $\Vert f{\Vert }_{\infty ,w}:={\mathrm{esssup}}_{u\in [-1,1]}\left(|f(u)|w(u)\right)$, ${\omega }_{\phi }^r(f,t{)}_{p,w}:={\sup }_{0<h\le t}\Vert {\Delta }_{h\phi (\cdot )}^r(f,\cdot ){\Vert }_{p,w}$, $E_n(f{)}_{p,w}:={\inf }_{P_n\in {\Pi }_n}\Vert f-P_n{\Vert }_{p,w}$, and ${\Pi }_n$ is the set of all algebraic polynomials of degree $\le n-1$. Equivalence type results involving related $K$-functionals and realization type results (obtained as corollaries of our estimates) are also discussed. Finally, we mention that (*) closes a gap left in the paper by G. Mastroianni and V. Totik "Best Approximation and moduli of smoothness for doubling weights", J. Approx. Theory {\bf 110} (2001), 180-199, where ($\ast$) was established for $p=\infty$ and ${\omega }_{\phi }^{r+2}$ instead of ${\omega }_{\phi }^{r+1}$ (it was shown there that, in general, ($\ast$) is not valid for $p=\infty$ if ${\omega }_{\phi }^{r+1}$ is replaced by ${\omega }_{\phi }^r$).