Rational approximation in $L^p$ and Faber transforms

Using the technique of Faber transforms we show that Pekarskii's theorem on rational approximation of functions in $H^p$, $1<p<\infty$, directly implies Petrushev's theorem on rational approximation in $L^p[-1,1]$, $1<p<\infty$, and vice versa. The same technique permits us to obtain similar results for functions analytic in domains with Lipschitz Jordan boundaries.