Rational approximation and smoothness of functions

For a compact set $K$ on the complex plane and a Banach space $X$ of functions on $K$ the numbers $r_n^X(f)$, $f\in X$, are defined by
$$ r_n^X(f)\overset{\text{def}}=\operatorname{int}_X\|f-r\|_X, $$
the infimum being taken over all rational functions $r=p/q$ where $\operatorname{deg}p\le n$, $\operatorname{deg}q\le n$ and $q$ does not vanisch on $K$. Thе question is to compare the smoothess of $f$ with the speed of decreasing of $r_n^X(f)$.
Two cases are considered: 1)$\operatorname{int}K$ a Jordan domain with Lipschitzian boundary,
$$ X=K^+_{L^\infty(\partial G)}\overset{\text{def}}= \biggl\{f:f(z)=\frac1{2\pi i}\int_{\partial G}\frac{g(\zeta)}{\zeta-z}d\zeta,\quad g\in L^\infty(\partial G)\biggr\}; $$
2)$K=[-1,1]$, $X=\mathrm{BMO}[-1,1]$. It is proved that $\sum_n(r_n^X)^p<+\infty$ if and only if $f$ belongs to the Besov class $B_p^{1/p}$.