# Orthonormalreihenentwicklungen für gewisse quasikonforme Normalabbildungen

### Erich Hoy

Friedberg, Germany

## Abstract

The paper deals with the construction of solutions for the equation $f_{zˉ}(z)=v(z)f_{z}(z) $ with $v(z)≡0$ in a finitely connected region $g$ and $v(z)≡q_{j}=$ const in the complementary continua $B_{j}$ of $G(0<q_{j}<1,j=1,2,…,n)$. The construction starts with well-known and in a simple way explicitly computable analytic functions in $G$, 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 $G$ of the square of derivative’s absolute value. If the boundary of $G$ consists of analytic Jordan curves only, then there is even an upper bound of the form $M∗ϱ∗_{m}$ with $M∗>0$ and $0<ϱ∗<1$ for the supremum of deviation of the $m$-th partial sum of these series from the sought solutions over $G$. Simple methods are given for the computation of $ϱ∗$.The results are generalized for the case, that in $B_{j}$ 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.

## Cite this article

Erich Hoy, Orthonormalreihenentwicklungen für gewisse quasikonforme Normalabbildungen. Z. Anal. Anwend. 3 (1984), no. 6, pp. 503–521

DOI 10.4171/ZAA/125