Does anyone know where to find results on the optimal polynomial approximability of functions such as $$ f(x):=\|x\|, x\in[-1,1]^d, $$ or $$ f(x):=\frac{1}{1+\|x\|}, x\in[-1,1]^d? $$
That is, for example for $ d=1$, I am interested in upper and lower bounds for $$ \inf_{p\in P_k}\|f-p\|_{L^p} $$ as $ k\to\infty $ ( $ p\in {2,\infty}$). Here $ P_k $ are polynomials of degree equal to or less than $ k $.
For $ d> 1$ I would be interested in similar results for any type of polynomial space you may offer.