Some properties for quaternionic slice regular functions on domains without real points

Altavilla, Amedeo

doi:10.1080/17476933.2014.889691

Cited by 27 publications

(66 citation statements)

References 10 publications

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“…As in the previous case, there exists a closed Euclidean ball K := B(x 0 , r) ⊆ U such that g never vanishes in K \ {x 0 } and we can prove along the same lines that f (U ) includes B(y 0 , ε) with ε := 1 3 min ∂K |g|. For quaternions, related results had been proven in [12,7,8,1] and more will appear in [15]. For octonions, the recent work [23] had considered the case of circular open subsets of a slice domain.…”

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confidence: 71%

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Division algebras of slice functions

Ghiloni¹,

Perotti²,

Stoppato³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

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confidence: 71%

“…The grounds for our work thus set, we proceed in Section 3 to a detailed description of the zero sets of slice functions over finite-dimensional division algebras. Their peculiar properties are direct extensions of those of quaternionic slice regular functions, [12,6,4,8,1], and of octonionic power series, [13,17].…”

Section: Introductionmentioning

confidence: 98%

Division algebras of slice functions

Ghiloni¹,

Perotti²,

Stoppato³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Nevertheless a different type of series expansion has been studied so far, namely the spherical power series [21,26,27], and it admits actual euclidean open domains as domains of convergence. For this reason the following proposition, already observed in the PhD thesis of the first author [4], holds true.…”

Section: Pql Functionsmentioning

confidence: 59%

Log-biharmonicity and a Jensen formula in the space of quaternions

Altavilla¹,

Bisi²

2019

Ann. Acad. Sci. Fenn. Math.

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Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generalization of these two concepts in the space of quaternions, obtaining new interesting Riesz measures and global (i.e. four dimensional), Jensen formulas.

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“…As in the case of slice regular functions, namely N = 1, we can introduce slice polyanalytic functions as a subclass of slice functions which are defined to be (see [10]): Then, we have the following: Inspired from the paper [5], we can show another version of the identity principle for slice polyanalytic functions without the hypothesis that the open set on which they are defined is a slice domain. First, note that slice functions can be recovered by their values on two semi-slices, see the Representation Formula given by Theorem 2.4 in [5]. We have the following Proof.…”

Section: Quaternionic Slice Polyanalytic Functionsmentioning

confidence: 99%

On Slice Polyanalytic Functions of a Quaternionic Variable

2019

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In this paper, we introduce the quaternionic slice polyanalytic functions and we prove some of their properties. Then, we apply the obtained results to begin the study of the quaternionic Fock and Bergman spaces in this new setting. In particular, we give explicit expressions of their reproducing kernels.Proposition 3.18. Let Ω be an axially symmetric domain and let f : Ω −→ H be a slice polyanalytic function. Assume that there exist J, K ∈ S, with J = K and U J , U K two subdomains of Ω + J and Ω + K respectively where Ω + J := Ω ∩ C + J and Ω + K := Ω ∩ C + K . If f = 0 on U J and U K , then f = 0 everywhere in Ω. Proof. Let f be a slice polyanalytic function on Ω such that f = 0 on U J and U K . Thus, since U J and U K are respectively subdomains of Ω + J and Ω + K . It follows, from the Splitting Lemma for slice polyanalytic functions combined with the classical complex analysis that f = 0 everywhere on Ω + J and Ω + K . Then, we just need to use the Representation Formula which allows to recover a slice function by its values on two semi-slices to complete the proof.

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Some properties for quaternionic slice regular functions on domains without real points

Cited by 27 publications

References 10 publications

Division algebras of slice functions

Division algebras of slice functions

Log-biharmonicity and a Jensen formula in the space of quaternions

On Slice Polyanalytic Functions of a Quaternionic Variable

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