Maximal convergence theorems for functions of squared modulus holomorphic type in $R^2$ and various applications
Authors
- Kraus, Christiane
2010 Mathematics Subject Classification
- 41A17 41A10 41A60 41A63 41A25 30E10 30C35
Keywords
- Polynomial approximation in 2-space, maximal convergence, Bernstein-Walsh's type theorems, real-analytic functions
DOI
Abstract
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.
Appeared in
- J. Approx. Theory, 147 (2007) pp. 47-66.
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