A New Theory of Regular Functions of a Quaternionic Variable

By Graziano Gentili, Daniele C. Struppa. 2007

473 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fea0d9a31b7f3483d3b3d8e6610f6f9546a1f23a

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In Algebra and Number Theory

In Applied Mathematics

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Citations in the corpus

The algebra of slice functions (2015, background): “In [11, 12], the authors constructed a class of quaternionic functions that includes the ring of polynomials just described (with R = H).”
A Cauchy kernel for slice regular functions (2009, methodology): “We will now pass to the case of slice regular functions, and we will recall the definition given by the authors in [13].”
Continuous slice functional calculus in quaternionic Hilbert spaces (2012, background): “Continuous slice functional calculus for normal operators 59 References 69”
The C-property for slice regular functions and applications to the Bergman space (2013, background): “After their definition in [15] which has been generalized to the case of functions with values in a Clifford algebra in [8], these functions have been intensively studied during the past years and their theory, as well as the application to the functional calculus, is treated in the book [9].”
The fundamental theorem of algebra for Hamilton and Cayley numbers (2008, background): “1 in [6] and [7]) f is the unique regular extension to K of the “complex” map fL I : L I → L I defined by f for w ∈ L I .”
M\"obius transformations and the Poincar\'e distance in the quaternionic setting (2008, background): “MÖBIUS TRANSFORMATIONS AND THE POINCARÉ DISTANCE ON H+ Similarly to what happens in the case of the complex plane, the quaternionic halfspace H+ = {q ∈ H : Re(q) > 0} is diffeomorphic to the open, unit disc ∆H via the (biregular) Cayley transformation ψ(q) = (1 + q)(1 − q)−1 ∈ G, (see [10]).”
Non Commutative Functional Calculus: Bounded Operators (2007, background): “In fact, every slice regular function can be represented as a power series, [3]: Theorem 2.”
Twistor transforms of quaternionic functions and orthogonal complex structures (2012, background): “In the present work, we solve such a problem working directly on H by exhibiting a rather surprising link with the class of quaternionic functions recently introduced in [15].”
Regular Functions of a Quaternionic Variable (2013, methodology): “Chapter 1 explains in detail the definition of regularity given by Gentili and Struppa in [29, 30].”
The Fueter mapping theorem in integral form and the ℱ‐functional calculus (2010, background): “Functions of this form are slice regular (in short s-regular) in the sense of [12].”
On the Multiplicity of Zeroes of Polynomials with Quaternionic Coefficients (2008, background): “Example 1.4. Consider the polynomial P2(q )=( q − α) ∗ (q − α )= q 2 − q(2Re(α)) + |α| 2 .I n this caseα is called a spherical root, see [ 3 ], and it is”
Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting (2011, background): “Some references for this theory of functions are [56, 33, 38].”
The open mapping theorem for regular quaternionic functions (2008, methodology): “As observed in [8], a quaternionic power series ∑ n∈N qan with an ∈ H defines a regular function in its domain of convergence, which proves to be an open ball B(0, R) = {q ∈ H : |q| < R}.”
Slice regular functions on real alternative algebras (2010, methodology): “A version of the Cauchy representation formula for quaternionic power series was proved in [19] and extended in [3].”
On Two Approaches to the Bergman Theory for Slice Regular Functions (2013, methodology): “In this paper we introduce the first elements and the settings of the Bergman theory in the case of slice regular functions studied in the papers [5–7, 9, 17].”
The inverse Fueter mapping theorem (2011, methodology): “The quaternionic version of the theory is based on the notion of slice regularity which was inspired by the paper of Cullen [16] and its further advances are in the papers [4], [18].”