Rational approximations of absolutely continuous functions with derivative in an Orlicz space

Let $R_n(f)$ be the best uniform approximation of $f \in C[0,1]$ by rational fractions of degree at most $n$, and let $ W[0,1]$ be the set of monotone convex functions $w\in C[0,1]$ such that $w(0)=0$ and $w(1)=1$.
Theorem 1. Suppose the function $f$ is absolutely continuous on the interval $[0,1],$ and let $w\in W[0,1]$ and $\widehat f= f(w(x))$. If $|\widehat f'|\ln^+|\widehat f'|$ is summable on $[0,1],$ then $R_n(f)=o(1/n)$.
Various applications and generalizations of this result are given, and the periodic case is also considered.
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