(11/18/2009, 10:43 AM)bo198214 Wrote: Yes, there you are right (and i was wrong). If I remember correctly you only need the Runge approximation for 1/(z-w) or so. So if you have the coefficients of the Runge approximation maybe you can deduce the Star-expansion coefficients following the proof.
Hmm. I was emailed a snippet of the 1977 book for Complex analysis (the one by Markushevich mentioned on the website), showing the missing other page of the proof you couldn't see in Google's book preview, with which had this:
(the comment in "[]"s is mine)
Quote:The function \( \frac{1}{1 - w} \) is single-valued and analytic on G [described as "the domain G bounded by the segment u >= 1, v = 0 (the part of the real axis going from 1 to \( \infty \)", which sounds like the Mittag-Leffler star of that function], and hence, by Runge's theorem (Theorem 3.5) there exists a sequence of polynomials
\( P_n(w) = c_0^{(n)} + c_1^{(n)} w + ... + c_{k_n}^{(n)} w^{k_n} \)
such that
\( lim_{n \rightarrow \infty} P_n(w) = \frac{1}{1 - w} \)
where the convergence is uniform inside G, in particular on the compact set \( E(F, L) \subset G \).
It mentions "Runge's theorem", does that have to do with "Runge approximations"? Can this be useful in determining the polynomials (and so the magic coefficients)? If so, how would one do it?
