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A general setting for functions of Fueter variables: differentiability, rational functions, Fock module and related topics

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We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field—assumed to be the real or the complex numbers—and which contains the field. Notably, we consider Fueter expansions, Gleason’s problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.

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Correspondence to Daniel Alpay.

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We thank the anonymous referee for their careful reading of this work and for their comments and suggestions.

Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research.

Ismael L. Paiva acknowledges financial support from the Science without Borders program (CNPq/Brazil).

Daniele C. Struppa thanks the Donald Bren Distinguished Chair in Mathematics, which supported this research.

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Alpay, D., Paiva, I.L. & Struppa, D.C. A general setting for functions of Fueter variables: differentiability, rational functions, Fock module and related topics. Isr. J. Math. 236, 207–246 (2020).

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