Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis

Alpay, Daniel; Luna-Elizarrarás, M. Elena; Shapiro, Michael; Struppa, Daniele C.

doi:10.1007/978-3-319-05110-9

Cited by 122 publications

(135 citation statements)

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“…Quite recently, there has been a significant resurgence of interest in the theory of functions of bicomplex variables, where some of these ideas are further expanded and discussed. Without any attempt to completeness, we should at least quote [3], [5], [11], [12], [13], [17] and [18].…”

Section: The Case N =mentioning

confidence: 99%

On Clifford analysis for holomorphic mappings

Luna-Elizarrarás¹,

Shapiro

Struppa

2014

Advances in Geometry

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In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n and m, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the CauchyRiemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

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Section: The Case N =mentioning

confidence: 99%

On Clifford analysis for holomorphic mappings

Luna-Elizarrarás¹,

Shapiro

Struppa

2014

Advances in Geometry

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“…The order defined in Theorem 3.1 was considered in [1] to introduce a new hyperbolic-valued norm on the set of bicomplex numbers.…”

Section: Archimedean F -Algebra Structure Theorem the Algebra Of Hypementioning

confidence: 99%

$${f}$$ f -Algebra Structure on Hyperbolic Numbers

Gargoubi

Kossentini

2016

Adv. Appl. Clifford Algebras

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The algebra of hyperbolic numbers is endowed with a partial order structure. We show that this system of numbers is the only (natural) generalization of real numbers into Archimedean f -algebra of dimension two. We establish various properties of hyperbolic numbers related to the f -algebra structure. In particular, we generalize fundamental properties of real numbers and give some order interpretations for the two dimensional space-time geometry.

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“…Elizarraràs et al [1,11] introduced elementary functions, such as polynomials, exponentials and trigonometric functions in this algebra, as well as their inverses, something that is not possible in the case of quaternions. They showed that any pair of holomorphic functions admits derivative in the sense of bicomplex numbers.…”

Section: Introductionmentioning

confidence: 99%