Orthogonalization in Clifford Hilbert modules and applications

Abstract

We prove that the Gram-Schmidt orthogonalization process can be carried out in Hilbert modules over Clifford algebras, in spite of the uninvertibility 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 ∈ 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 R m .

Adaptive Fourier decomposition‐type sparse representations versus the Karhunen–Loève expansion for decomposing stochastic processes

“…With such formulation, Core AFD is extended to contexts of great variety in which a practical Blaschke product theory may not be known or may not exist. Significant generalizations include POAFD for product dictionary [10], POAFD for quaternionic space [22], POAFD for multivariate real variables in the Clifford algebra setting [23],…”