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 \[ E_n(f)_{ p, w_n} = O(n^{ -\alpha}) \iff \omega_\varphi^{ r+1}(f, n^{ -1})_{ p, w_n} = O(n^{ -\alpha}), \] where $\lambda_p := p$ if $0 < p < \infty$, $\lambda_p :=1$ if $p=\infty$, $\|f\|_{ p,w} := \left( \int_{ -1}^1 |f(u)|^p w(u) du \right)^{ 1/p}$, $\|f\|_{ \infty,w} := { \mathop{ \rm ess\: sup}\nolimits}_{ u\in [-1,1]} \left(|f(u)| w(u)\right)$, $\omega_\varphi^r(f, t)_{ p, w} := \sup_{ 0<h\leq t} \| \Delta_{ h\varphi(\cdot)}^r(f,\cdot)\|_{ p, w}$, $E_n(f)_{ p, w} := \inf_{ P_n\in\Pi_n} \|f-P_n\|_{ p,w}$, and $\Pi_n$ is the set of all algebraic polynomials of degree $\leq n-1$. 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 $\omega_\varphi^{ r+2}$ instead of $\omega_\varphi^{ r+1}$ (it was shown there that, in general, ($\ast$) is not valid for $p=\infty$ if $\omega_\varphi^{ r+1}$ is replaced by $\omega_\varphi^{ r}$).
Categories: math.CA
