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 to 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 in \({\mathbb {R}}^m\) and on the sphere.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11701105), Research Grant of Macau University of Science and Technology (No. FRG-22-075-MCMS) and Macao Government Research Funding (No. FDCT0128/2022/A). We thank the reviewers for their careful reading of the manuscript and valuable comments and suggestions.
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Communicated by Deguang Han.
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Wang, J., Qian, T. Orthogonalization in Clifford Hilbert modules and applications. Banach J. Math. Anal. 18, 2 (2024). https://doi.org/10.1007/s43037-023-00312-y
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DOI: https://doi.org/10.1007/s43037-023-00312-y