Coefficients of lacunary series on quasiconformally transformed unit disk - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:10:15Z http://mathoverflow.net/feeds/question/65460 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65460/coefficients-of-lacunary-series-on-quasiconformally-transformed-unit-disk Coefficients of lacunary series on quasiconformally transformed unit disk deoxygerbe 2011-05-19T19:08:11Z 2011-06-25T18:22:12Z <p>Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if $|q|=1$ then $|\xi(q)|=1$. Is there a method from Teichmüller theory that allows us to explicitly write down the coefficients $b_{n}$ of $s(\xi(q)) = \sum_{n=0}^{\infty} a_{n}\xi(q)^{n} = \sum_{n=0}^{\infty} b_{n}q^{n}$ given some explicit $\xi$?</p> http://mathoverflow.net/questions/65460/coefficients-of-lacunary-series-on-quasiconformally-transformed-unit-disk/67523#67523 Answer by Sylvain Bonnot for Coefficients of lacunary series on quasiconformally transformed unit disk Sylvain Bonnot 2011-06-11T16:52:37Z 2011-06-11T16:52:37Z <p>The answer is no: pick $s(q)=q$ to be simply the identity, then you would obtain automatically a power series expansion for any quasiconformal $\xi$ preserving the circle, which can't be true since there are some $\xi$ that are not analytic near the origin...</p>