Holomorphic approximation of \(L_2\)-functions on the unit sphere in \(\mathbb R^3\)
type: Item
instance of: scholarly article
title: Holomorphic approximation of \(L_2\)-functions on the unit sphere in \(\mathbb R^3\)
zbMATH Open document ID: 1312.30063
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-26T00:00:00Z
review text: The authors note, first of all, that the most widely used approximation for \(L_{2}\)-functions on the unit sphere in \(\mathbb{R}^{m} \) is the approximation by spherical harmonics, that is, if \(S_{k}(\underline{x})\) is a \( k \)-homogeneous harmonic polynomial then every \(f\in L_{2}\left(S^{m-1}\right)\) admits an orthogonal decomposition in spherical harmonics of the form \[ f(\underline{\omega})=\Sigma_{k=0}^{\infty}S_k(\underline{\omega}),\quad \underline{\omega}=\frac{\underline{x}}{\left|{\underline{x}}\right|}\in S^{m-1}. \] A similar situation occurs when monogenic functions of Clifford analysis are used instead of harmonic functions; here for any scalar function \(f\in L_{2}\left(S^{m-1}\right)\) there exists a series of \(k\)-homogeneous monogenic polynomials (called monogenic signals) such that \(f\) coincides with the scalar part of the series. But the product of monogenic functions is no longer monogenic which is a kind of disadvantage, and the paper deals with the idea of constructing an embedding of holomorphic functions of several complex variables into the unit ball in Euclidean space. Let \(e_{1}\) and \(e_{2}\) be Clifford generators of the skew-field of quaternions \(\mathbb{H}\), then the ``holomorphic correspondence'' is the map from the holomorphic functions in two complex variables \( \displaystyle g\left(z_{1},z_{2}\right)=\Sigma_{p,q=0}^{\infty}z_{1}^{p}z_{2}^{q}C_{p,q}\) into the corresponding series in \(\mathbb{R}^{3}\): \[ g\left(z_{1},z_{2}\right)\mapsto \Sigma_{p,q=0}^{\infty}\left(x_{0}+\underline{x}\right)^{p}\left(x_{1}-e_{1}e_{2}x_{2}\right)^{q}C_{p,q} \] with \(\underline{x}=x_{1}e_{1}+x_{2}e_{2}\). This embedding leads to a closed subspace of \(L_{2}\left(S^{2}\right)\) called the space holomorphic signals on \(S^{2}\). After the introductory material in Section 1 and some preliminaries in Section 2, one finds in Section 3 the construction of the Gram-Schmidt orthonormal basis that corresponds to the holomorphic polynomials …
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 6390674
zbMATH Keywords: quaternionic analysis
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: holomorphic signals
zbMATH Keywords: holomorphic polynomials
reviewed by: M. Shapiro
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.02.065
OpenAlex ID: W203184771
cites work: Gelfand-Tsetlin bases for spherical monogenics in dimension 3
cites work: Q3310867
cites work: Axial monogenic functions from holomorphic functions
cites work: Q4003806
cites work: Q3995205
cites work: Q3671491
cites work: Q4382291
cites work: Q5535148
cites work: Q5518575
cites work: A product and an exponential function in hypercomplex function theory
cites work: Special functions in Clifford analysis and axial symmetry
DOI: 10.1016/J.JMAA.2014.02.065
description: scientific article; zbMATH DE number 6390674