Slice regular functions on real alternative algebras

By Riccardo Ghiloni, A. Perotti. 2010

252 citations recorded by the source corpus.

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well–known Fueter’s construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C 1 .

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/efc111be13ceb3045f599de96a64273b93ef1a59

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Citations in the corpus

A new series expansion for slice regular functions (2011, background): “This last result that has been improved to f ∈ C(Ω) in [13].”
Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting (2011, unknown): “For the generalization to functions with values in a Clifford algebra, called again slice hyperholomorphic or slice monogenic functions, we refer the reader to [44, 46, 45], and to [57] for functions with values in a real alternative algebra.”
Continuous slice functional calculus in quaternionic Hilbert spaces (2012, background): “Continuous slice functional calculus for normal operators 59 References 69”
Global differential equations for slice regular functions (2014, methodology): “By combining the latter fact with Proposition 7,(2) of [7], we infer that f1 is continuous.”
Entire slice regular functions (2015, background): “The theory was developed also for functions with values in a Clifford algebra, see [72], [73], for functions with values in a real alternative algebra, see [117].”
A nonconstant coefficients differential operator associated to slice monogenic functions (2012, background): “However, there is a class of domains on which the two sets of functions coincide, as proven in [5], [15].”
The algebra of slice functions (2015, methodology): “The new approach introduced in [16] makes it possible to address the case of other alternative -algebras A and to generalize part of the results to slice functions that are not regular.”