Abstract:
The author establishes that, for every function $f(z)$ that is analytic inside the unit disk $D$ and belongs to the space $L^p(D)$ with $p>1$, the equation
$$
\rho\stackrel{\operatorname{def}}{=}\varlimsup_{n\to\infty}\sqrt[\leftroot{2}\uproot{4}n]{L^pE_n(f,D)-L^pR_n(f,D)}=\varlimsup_{n\to\infty}\sqrt[\leftroot{2}\uproot{4}n]{L^pE_n(f,D)}
$$
is satisfied, where $L^pE_n(f,D)$ and $L^pR_n(f,D)$ are the minimal deviations of $f$ from polynomials of degree at most $n$ and from rational functions of order at most $n$. In particular, $\rho<1$ if and only if $f$ can be continued analytically over the disk $|z|<1/\rho$.
There is also a similar proposition for the approximation of functions in the spaces $H^p$,
$p>1$.
Fulltext:
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