Orthogonalization in Clifford Hilbert modules and applications

TL;DR

This paper extends the Gram--Schmidt orthogonalization process to Clifford Hilbert modules and applies it to construct orthonormal bases, define Clifford rational systems, and develop an adaptive approximation theory for functions on spheres and in Euclidean space.

Contribution

It introduces a method to perform orthogonalization in Clifford modules and applies it to construct bases and rational systems for approximation theory.

Findings

Constructed orthonormal bases of inner spherical monogenics.

Formulated Clifford Takenaka--Malmquist systems and Blaschke products.

Established an adaptive rational approximation theory for L^2 functions.

Abstract

We prove that the Gram--Schmidt orthogonalization process can be carried out in Hilbert modules over Clifford algebras, in spite of the un-invertibility and the un-commutativity of general Clifford numbers. Then we give two crucial applications of the orthogonalization method. One is to give a constructive proof of existence of an orthonormal basis of the inner spherical monogenics of order $k$ for each $k\in\mathbb{N}.$ The second is to formulate the Clifford Takenaka--Malmquist systems, or in other words, the Clifford rational orthogonal systems, as well as define Clifford Blaschke product functions, in both the unit ball and the half space contexts. The Clifford TM systems then are further used to establish an adaptive rational approximation theory for $L^2$ functions on the sphere and in $\mathbb{R}^m.$