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Keywords = special monogenic polynomials

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20 pages, 346 KiB  
Article
Equivalent Base Expansions in the Space of Cliffordian Functions
by Mohra Zayed and Gamal Hassan
Axioms 2023, 12(6), 544; https://doi.org/10.3390/axioms12060544 - 31 May 2023
Cited by 3 | Viewed by 1181
Abstract
Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. [...] Read more.
Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. In this paper, first, we derive a new base of special monogenic polynomials (SMPs) in Fréchet–Cliffordian modules, named the equivalent base, and examine its convergence properties for several cases according to certain conditions applied to related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions, such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined and proved to be attainable. This work improves and generalizes several existing results in the complex and Clifford context involving the convergence properties of the product and similar bases. Full article
(This article belongs to the Special Issue Recent Advances in Complex Analysis and Applications)
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