Exponential Bases of Cliffordian Polynomials in Fréchet Modules

Abstract

This paper presents a generalization of the exponential base of special monogenic polynomials within the framework of Fr ́echet modules (F-modules). The study focuses on examining the convergence properties, specifically the effectiveness, of both the exponential simple base of special monogenic polynomials (ESBSMPs) and the exponential Cannon base of special monogenic
polynomials (ECBSMPs) in Fr ́echet modules. These properties are investigated on hyper-closed and open balls, in open regions surrounding hyper-closed balls, for all entire special monogenic functions, as well as at the origin. Furthermore, an explicit upper bound for the order of the exponential simple base is established and shown to be attainable. Finally, we extend the discussion to equivalent and similar bases, verifying that the derived results remain valid under such transformations, which confirms the robustness and general applicability of the findings.