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Beschreibung:
In this paper we extend the theory of maximal convergence introduced by Walsh to functions of squared modulus holomorphic type. We introduce in accordance to the well-known complex maximal convergence number for holomorphic functions a real maximal convergence number for functions of squared modulus holomorphic type and prove several maximal convergence theorems. We achieve that the real maximal convergence number for F is always greater or equal than the complex maximal convergence number for g and equality occurs if L is a closed disk in R^2. Among other various applications of the resulting approximation estimates we show that for functions F of squared holomorphic type which have no zeros in a closed disk B_r the relation $$ \limsup_{n \to \infty} \sqrt[n]{ E_n( B_r,F)} = \limsup_{n \to \infty} \sqrt[n]{E_n( { \partial B}_r,F)} $$ is valid, where E_n is the polynomial approximation error.