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Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. By Riemann mapping theorem, there exists a unique exterior conformal representation $\Phi$ from $\overline{\mathbb{C}}\setminus E$ onto $\overline{\mathbb{C}}\setminus \{w: |w|\leq 1\}$ satisfying $\Phi(\infty)=\infty$ and $\Phi'(\infty)>0.$ Define $$\Psi(w):=\Phi^{-1}(w),$$ for any $w\in \overline{\mathbb{C}}\setminus \{w: |w|\leq 1\}.$ Let $F$ be a holomorphic function on a neighborhood of $E.$ Define $$f(w):=F(\Psi(w)),$$ when $w\in \{w: 1<|w|\leq r\}$ for some $r>1.$

I would like to have your help about the proof of the following statement:

If the regular part (around $0$) of $f(w)$ is a rational function with at most $m-1$ poles, then $F(z)$ is a rational function with at most $m-1$ poles.

This statement is in a paper without proof and I have no idea how to prove it. Could someone provide me a proof of the statement or give me an idea of the proof? Thank you very much.

Masik

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  • $\begingroup$ Your function $f$ is not defined in the whole disk $|z|>1$, only in a ring $1<|z|<r$ for some $r$, because $F$ is defined only in a neighborhood of $E$. So what do you mean by "regular part at $0$"? $\endgroup$ – Alexandre Eremenko Jan 13 '18 at 13:22
  • $\begingroup$ Thank you very much for your comment. You are totally right! $f$ should be defined only in a ring $1<|w|<r$ for some $r.$ My question is "Can we prove that if the regular part (around $0$) of $f(w)$ is a rational function with at most $m-1$ poles, then $F(z)$ is a rational function with at most $m-1$ poles?" $\endgroup$ – Masik Kara Jan 13 '18 at 16:16
  • $\begingroup$ To be precise, we assume that the regular part (around $0$) of $f(w)$ is a rational function with at most $m-1$ poles. We want to prove that $F(z)$ is a rational function with at most $m-1$ poles. I found this statement in the paper "An analogue of Fabry's theorem for generalized Padé approximants" without any proof. I would like to know how to prove it. $\endgroup$ – Masik Kara Jan 13 '18 at 16:23
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Yes, this is correct. Suppose that $$f(w)=F\circ\Psi(w)=R(w)+S(w),\quad 1<|w|<r$$ where $R$ is the regular part and $S$ is the singular part, and $r>1$. The singular part is convergent in $\Delta:=\overline{C}\backslash\{ z:|z|\leq 1\}$ (by definition of Laurent series) and $R$ is rational, so $f$ is meromorphic in $\Delta$. If $R$ has $m$ poles, then $f$ has $\leq m$ poles in $\Delta$. As $\Psi$ maps $\Delta$ onto $\overline{C}\backslash E$, biholomorphically, we conclude that $F=f\circ\Phi,\; \Phi=\Psi^{-1}$ has a meromorphic continuation to $\overline{C}\backslash E$, with $\leq m$ poles. This proves the statement, as every meromorphic function on the Riemann sphere is rational.

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