Gleason's problem and interpolation for hyperholomorphic functions).
type: Item
instance of: scholarly article
title: Gleason's problem and interpolation for hyperholomorphic functions).
zbMATH Open document ID: 1039.30026
DOI: 10.1016/S1631-073X(02)02598-0
author: Daniel Alpay
author: M. Shapiro
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-05-27T00:00:00Z
review text: The authors generalize the following Gleason problem: Given a family \({\mathcal F}\) of analytic functions in \(\Omega\subset \mathbb{C}^N\) and given a point \(a\in{\mathcal F}\) such that: \[ f(z)- f(a)= \sum^N_{i=1} (z_i- a_i) g_i(z).\tag{1} \] Let \(N= 2\) and \(\Omega\subset\mathbb{C}^2= \mathbb{H}\). In (1) we consider now functions \(g_i\) which are left hyperholomorhic with quaternionic coefficients. For \(a= 0\) the functions \[ g_i(x):= \int^1_0 \partial_i f(tx)\,dt \] defined in the unit ball \(S\) are also left hyperholomorphic. The proof is obtained using Leibenson's method. For an arbitrary point \(a\in S\) the representation \[ f(x)- (1-\overline a x){(1-| a|^2)^3\over| 1-\overline ax|^4} f(a)= \zeta(x,a) \begin{pmatrix} g_1(x)\\ g_2(x)\\ g_3(x)\end{pmatrix}, \] is obtained where \(\zeta: S\to \mathbb{H}^3\) has to exist and \(g_i\), \(f\) are left hyperholomorphic functions. This result is interpreted in pairs of complex functions. Finally, the homogeneous interpolation problem in the setting of left hyperholomorphic functions can be studied.
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 1921082
zbMATH Keywords: Gleason's problem
zbMATH Keywords: hyperholomorphic function
MaRDI profile type: Publication
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full work available at URL: s1631-073x(02)02598-0
OpenAlex ID: W2018930703
description: scientific article; zbMATH DE number 1921082