WIAS Preprint No. 1175, (2006)

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

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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