2021/03/17 by Jinxun Wang, Wang, Jinxun, Tao Qian +1
Mathematics · #15A66 #30G35 #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2103.09416
openalex publication_date 2021/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∈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 L2 functions on the sphere and in ℝm.