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Analytic functions and analytic functionals on some balls in the complex Euclidean space. (English) Zbl 1045.32001
Begehr, Heinrich G. W. (ed.) et al., Analysis and applications–ISAAC 2001.
Proceedings of the 3rd international congress, Berlin, Germany, August 20–25, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1384-1/hbk). Int. Soc. Anal. Appl. Comput. 10, 151-159 (2003).
Generalizing the Lie norm, the Euclidean norm and the dual Lie norm, the authors define a series of norms \(\{N_p\}_{1\leq p\leq\infty}\) on \(\mathbb C^{n+1}\), consider holomorphic functions, entire functions of exponential type and analytic functionals on the \(N_p\)-balls \(\widetilde{B}_p(r)\), and characterize them by their growth behavior of their harmonic components in their double series expansion. By means of these results, the Martineau’s theorem on Fourier-Borel transform is proved in the case of \(N_p\)-norm on the double series expansion.
For the entire collection see [Zbl 1031.35002].
32A10 Holomorphic functions of several complex variables
32A05 Power series, series of functions of several complex variables