Holomorphic approximation of \(L_2\)-functions on the unit sphere in \(\mathbb R^3\) (Q488750)

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 \(\left(x_{0}+\underline{x}\right)^{p}\left(x_{1}-e_{1}e_{2}x_{2}\right)^{q}\). In Section 4 the convergence properties for these holomorphic bases are studied, and for a fixed \(q\) the series converges for all \(x_{0}+\underline{x}\) with \(x_{0}^{2}+|\underline{x}|^{2}< 1\). In case both \(p\) and \(q\) are variable, the convergence is proved for \(\sqrt{x_{0}^{2}+|\underline{x}|^{2}}<\sqrt{2}-1\).