I'm referring to the article of N. Nadirashvili "Hadamard's and Calabi-Yau conjectures on negatively curved and minimal surfaces". In the proof of proposition 4.3 author use a theorem of Walsh. Now the only theorem of Walsh that i know is the well known theorem of equiconvergence (involving lagrange interpolation polynomials). I have two questions:
- Does Nadirashvili use the theorem of Walsh cited above in proposition 4.3? (if the answer is yes i don't understand how he use it)
- If he use another theorem of Walsh what is this theorem? Thank you
I think Nadirashvili blends two results of Walsh, one from 1926, another from 1928. (Neither of them is about equiconvergence).
Let's see what is going on. We have a function $$ \frac{1}{d} = \frac{1}{w'-g'/g} = \frac{q}{qw'-1} $$ which is holomorphic on the closure of the Jordan domain $E$. Notice that when $q$ vanishes, so does $1/d$. Moreover, if $q=c_j(z-z_j)^{k_j}+\dots$ near the zero $z_j$, then $\frac{1}{d}=-c_j(z-z_j)^{k_j}+\dots$.
Our goal is to approximate $1/d$ with a polynomial $s_\delta$ that has the same asymptotic behavior at the points $z_j$: this is what Nadirashvili expresses by $|s_\delta(z)+q(z)|=o(|z-z_j|^{k_j})$.
The natural thing to do is to factor our the zeros: the function $$F(z)=\frac{1}{d(z)}\prod_{j=1}^n (z-z_j)^{-k_j}$$ is also holomorphic on $\overline{E}$. We can approximate $F$ by a polynomial $P$ according to a theorem of Walsh, which is now a special case of Mergelyan's theorem, but came much earlier:
Source: Joseph L. Walsh: Selected Papers edited by Theodore J. Rivlin and E. B. Saff.
But this isn't enough, because the difference between $1/d$ and the product $F(z)\prod_{j=1}^n (z-z_j)^{k_j}$ can be as large as $O(|z-z_j|^{k_j})$, instead of $o(|z-z_j|^{k_j})$. What we need is to approximate $F$ by a polynomial that takes the same values as $F$ at the points $z_1,\dots,z_n$. This is another theorem of Walsh, which I take from the book by Walsh that Nadirashvili cites:
Note that this theorem applies to any compact set, as long as uniform approximation is possible (the characterization of such compact sets was not available until Mergelyan's theorem). The original paper in which this result is proved is in free access, see Theorem X.