2011
DOI: 10.1090/s0002-9939-2010-10812-2
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The Runge theorem for slice hyperholomorphic functions

Abstract: Abstract. In this paper we introduce and study rational slice monogenic functions. After proving a decomposition theorem for such functions, we are able to prove the Runge approximation theorem for slice monogenic functions. We then show how a similar argument can be used to obtain an analogue of the Runge approximation theorem in the slice regular setting.

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Cited by 28 publications
(27 citation statements)
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“…Indeed, the equality f I * f c I = f c I * f I (which is equivalent to N (f ) = N (f c ), in view of Remark 2.11 (v) and the identity principle) is used without proving it. A different approach has been used in [10,Proposition 3.2]. A complete treatment has been given in [22,Section 2] in the case of slice functions, which subsumes the case of slice monogenic functions.…”
Section: Preliminariesmentioning
confidence: 99%
“…Indeed, the equality f I * f c I = f c I * f I (which is equivalent to N (f ) = N (f c ), in view of Remark 2.11 (v) and the identity principle) is used without proving it. A different approach has been used in [10,Proposition 3.2]. A complete treatment has been given in [22,Section 2] in the case of slice functions, which subsumes the case of slice monogenic functions.…”
Section: Preliminariesmentioning
confidence: 99%
“…Recently, Gentili and Struppa proposed a different approach, which led to a new notion of holomorphicity (called slice regularity) for quaternion-valued functions of a quaternionic variable [15,16]. Unlike Fueter's, this theory includes the polynomials and the power series of the quaternionic variable q of the type n≥0 q n a n , with coefficients a n ∈ H. Furthermore, the analogs (sometime peculiarly different) of many of the fundamental properties of holomorphic functions of one complex variable can be proven in this new setting, like the Cauchy and Pompeiu representation formulas and Cauchy inequalities, the maximum (and minimum) modulus principle, the identity principle, the open mapping theorem, the Morera theorem, the power and Laurent series expansion, the Runge approximation theorem, to cite only some of the most significant (see [4][5][6][7][12][13][14][15][16]24]). In fact, the theory of slice regular functions is already rather rich and well established on steady foundations, and appears to be of fundamental importance to construct a functional calculus in non commutative settings [8].…”
Section: Introductionmentioning
confidence: 99%
“…In order to prove the vanishing of the first cohomology group with coefficients in the sheaf \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {SR}$\end{document} we need Runge's theorem. We state two versions of this result and we refer the reader to 14 for their proofs.…”
Section: Mittag‐leffler Theoremmentioning
confidence: 99%
“…Generally speaking, properties which hold for slice monogenic functions, in the sense that they have values in a Clifford algebra, can be stated for slice regular functions. The converse is not always true, but in this case the only key tool in the proofs is the Runge theorem, which the authors prove for the slice monogenic case in 14.…”
Section: Mittag‐leffler Theoremmentioning
confidence: 99%
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