Orthonormalreihenentwicklungen für gewisse quasikonforme Normalabbildungen

Abstract

The paper deals with the construction of solutions for the equation $f_{\bar{z}}(z)=v(z)\overline{f_z(z)}$ with $v(z)\equiv 0$ in a finitely connected region $\mathfrak{g}$ and $v(z)\equiv q_j=$ const in the complementary continua ${\mathfrak{B}}_j$ of $\mathfrak{G}(0<q_j<1,j=1,2,\dots ,n)$. The construction starts with well-known and in a simple way explicitly computable analytic functions in $\mathfrak{G}$, and series for the solutions are received only by the use of orthogonalization processes. These series converge in the well-known norm produced by the integral over $\mathfrak{G}$ of the square of derivative’s absolute value. If the boundary of $\mathfrak{G}$ consists of analytic Jordan curves only, then there is even an upper bound of the form $M\ast \varrho {\ast }^m$ with $M\ast >0$ and $0<\varrho \ast <1$ for the supremum of deviation of the $m$-th partial sum of these series from the sought solutions over $\mathfrak{G}$. Simple methods are given for the computation of $\varrho \ast$.The results are generalized for the case, that in ${\mathfrak{B}}_j$ analytic functions take the place of the constants $q_j$. At the conclusion a possible extension of the procedure to more generalized functions $v(z)$ is discussed.