The Weierstrass factorization theorem for slice regular functions over the quaternions

Gentili, Graziano; Vignozzi, Irene

doi:10.1007/s10455-011-9266-0

Cited by 16 publications

(22 citation statements)

References 21 publications

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“…whence |h(z)| ≤ c|z| < cR. By Proposition 6.10 of [21], we conclude that |h(q)| < 2cR, as desired. then |∂ c f (p)| = 1 2ρ0 and we have, for all z ∈ B I (0, ρ 1 ) ⊃ B I (p, ρ 0 ),…”

Section: A New Approach To the Quaternionic Bloch-landau Theoremsupporting

confidence: 53%

Landau’s theorem for slice regular functions on the quaternionic unit ball

Bisi

Stoppato

2017

Int. J. Math.

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Along with the development of the theory of slice regular functions over the real algebra of quaternions H during the last decade, some natural questions arose about slice regular functions on the open unit ball B in H. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of B fixing the origin, it establishes two variants of the quaternionic Schwarz-Pick lemma, specialized to maps B → B that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps f of the complex unit disk with f (0) = 0. Landau had computed, in terms of a := |f ′ (0)|, a radius ρ such that f is injective at least in the disk ∆(0, ρ) and such that the inclusion f (∆(0, ρ)) ⊇ ∆(0, ρ 2 ) holds. The analogous result proven here for slice regular functions B → B allows a new approach to the study of Bloch-Landau-type properties of slice regular functions B → H.

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Section: A New Approach To the Quaternionic Bloch-landau Theoremsupporting

confidence: 53%

Landau’s theorem for slice regular functions on the quaternionic unit ball

Bisi

Stoppato

2017

Int. J. Math.

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“…Weierstrass theorem was original proved in [113]. Here we provide an alternative statement and we will show how to retrieve the result in [113].…”

Section: Weierstrass Theoremmentioning

confidence: 90%

“…Function theory. The theory of slice regular functions was delevoped in the papers [47,104,106,107,109,110,112,113], in particular, the zeros were treated in [103,105,111] while further properties can be found in [33,44,85,92,93,102,140,141,150,151,152,154]. Slice monogenic functions with values in a Clifford algebra and their main properties were studied in [58,72,73,74,75,78,81,122,155].…”

Section: Introductionmentioning

confidence: 99%

Entire Slice Regular Functions

Colombo

Sabadini

Struppa

2016

SpringerBriefs in Mathematics

112

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The notion of supershift (in itself a generalization of the notion of superoscillation arising in quantum mechanics) expresses the fact that the sampling of a function in an interval allows to compute the values of the function far from the interval. In this paper we study the relation between supershift and real analyticity. We use a classical result due to Serge Bernstein to show that real analyticity for a complex valued function implies a strong form of supershift. On the other hand, we use a parametric version of a result by Leonid Kantorovitch to show that the converse is not true. We additionally study the stability of functions that satisfy suitable supershift requirements under multiplication by complex numbers or primitivization.

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“…Since the seminal paper by Gentili and Struppa [12], several articles [4,5,10,11,14,15,16,18,20,22,23] and some monographs [7,8,14] have been published in the field of (quaternionic) slice regular functions. The theory was mainly built to allow quaternionic polynomials to be regular and to mime, in some sense, the theory of complex holomorphic functions.…”

Section: Introductionmentioning

confidence: 99%

s-Regular functions which preserve a complex slice

Altavilla

Fabritiis

2018

Annali di Matematica

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We study global properties of quaternionic slice regular functions (also called s-regular ) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a "Hermitian" product on slice regular functions which gives us the possibility to express the * -product of two s-regular functions in terms of the scalar product of suitable functions constructed starting from f and g. Afterwards we are able to determine, under different assumptions, when the sum, the * -product and the * -conjugation of two slice regular functions preserve a complex slice. We also study when the * -power of a slice regular function has this property or when it preserves all complex slices. To obtain these results we prove two factorization theorems: in the first one, we are able to split a slice regular function into the product of two functions: one keeping track of the zeroes and the other which is never-vanishing; in the other one we give necessary and sufficient conditions for a slice regular function (which preserves all complex slices) to be the symmetrized of a suitable slice regular one.

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The Weierstrass factorization theorem for slice regular functions over the quaternions

Cited by 16 publications

References 21 publications

Landau’s theorem for slice regular functions on the quaternionic unit ball

Landau’s theorem for slice regular functions on the quaternionic unit ball

Entire Slice Regular Functions

s-Regular functions which preserve a complex slice

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