Polynomial approximation with doubling weights

Among other things, we prove that, for a doubling weight $w$, $0< p\leq\infty$,$r\in{\mathbb N}0$,and$0<\alpha <r+1 - 1/\lambda_p$, we have $$E_n(f{)}_{p,w_n}=O(n^{-\alpha })⟺{\omega }_{\phi }^{r+1}(f,n^{-1}{)}_{p,w_n}=O(n^{-\alpha }),$$ where ${\lambda }_p:=p$ if $0<p<\mathrm{\infty }$, ${\lambda }_p:=1$ if $p=\mathrm{\infty }$, $|f|{p,w} := \left( \int_{-1}^1 |f(u)|^p w(u) du \right)^{1/p}$,$|f|{\infty,w} := {\mathop{\rm ess: sup}\nolimits}{u\in [-1,1]} \left(|f(u)| w(u)\right)$,$\omega_\varphi^r(f, t){p, w} := \sup{0<h\leq t} | \Delta_{h\varphi(\cdot)}^r(f,\cdot)|{p, w}$,$E_n(f){p, w} := \inf_{P_n\in\Pi_n} |f-P_n|_{p,w}$,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 $p=\mathrm{\infty }$ and ${\omega }_{\phi }^{r+2}$ instead of ${\omega }_{\phi }^{r+1}$ (it was shown there that, in general, ($\ast$) is not valid for $p=\mathrm{\infty }$ if ${\omega }_{\phi }^{r+1}$ is replaced by ${\omega }_{\phi }^r$).