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# Zeitschrift für Analysis und ihre Anwendungen

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Volume 3, Issue 6, 1984, pp. 503–521
DOI: 10.4171/ZAA/125

Published online: 1984-12-31

Orthonormalreihenentwicklungen für gewisse quasikonforme Normalabbildungen

Erich Hoy

(1) Friedberg, Germany

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.