The Runge theorem for slice hyperholomorphic functions
HTML articles powered by AMS MathViewer
- by Fabrizio Colombo, Irene Sabadini and Daniele C. Struppa
- Proc. Amer. Math. Soc. 139 (2011), 1787-1803
- DOI: https://doi.org/10.1090/S0002-9939-2010-10812-2
- Published electronically: December 13, 2010
- PDF | Request permission
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.References
- F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
- Fabrizio Colombo, Graziano Gentili, and Irene Sabadini, A Cauchy kernel for slice regular functions, Ann. Global Anal. Geom. 37 (2010), no. 4, 361–378. MR 2601496, DOI 10.1007/s10455-009-9191-7
- F. Colombo, G. Gentili, I. Sabadini, D.C. Struppa, Non commutative functional calculus: Bounded operators, Complex Analysis and Operator Theory, 4 (2010), 821–843.
- Fabrizio Colombo, Graziano Gentili, Irene Sabadini, and Daniele C. Struppa, Non-commutative functional calculus: unbounded operators, J. Geom. Phys. 60 (2010), no. 2, 251–259. MR 2587392, DOI 10.1016/j.geomphys.2009.09.011
- Fabrizio Colombo and Irene Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009), no. 3, 601–627. MR 2496568, DOI 10.1007/s12220-009-9075-x
- Fabrizio Colombo, Irene Sabadini, Franciscus Sommen, and Daniele C. Struppa, Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, vol. 39, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2089988, DOI 10.1007/978-0-8176-8166-1
- Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008), no. 8, 2255–2274. MR 2402108, DOI 10.1016/j.jfa.2007.12.008
- Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009), 385–403. MR 2520116, DOI 10.1007/s11856-009-0055-4
- Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009), 385–403. MR 2520116, DOI 10.1007/s11856-009-0055-4
- Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, Duality theorems for slice hyperholomorphic functions, J. Reine Angew. Math. 645 (2010), 85–105. MR 2673423, DOI 10.1515/CRELLE.2010.060
- Fabrizio Colombo, Graziano Gentili, Irene Sabadini, and Daniele Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math. 222 (2009), no. 5, 1793–1808. MR 2555912, DOI 10.1016/j.aim.2009.06.015
- F. Colombo, I. Sabadini, A structure formula for slice monogenic functions and some of its consequences, Hypercomplex Analysis, Trends in Mathematics, Birkhäuser, 2009, 101–114.
- F. Colombo, I. Sabadini, D. C. Struppa, The Pompeiu formula for slice hyperholomorphic functions, to appear in Michigan J. Math. (2011).
- C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J. 32 (1965), 139–148. MR 173012
- Run Fueter, Die Funktionentheorie der Differentialgleichungen $\Theta u=0$ und $\Theta \Theta u=0$ mit vier reellen Variablen, Comment. Math. Helv. 7 (1934), no. 1, 307–330 (German). MR 1509515, DOI 10.1007/BF01292723
- Graziano Gentili and Daniele C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable, C. R. Math. Acad. Sci. Paris 342 (2006), no. 10, 741–744 (English, with English and French summaries). MR 2227751, DOI 10.1016/j.crma.2006.03.015
- Graziano Gentili and Daniele C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), no. 1, 279–301. MR 2353257, DOI 10.1016/j.aim.2007.05.010
- R. Ghiloni, A. Perotti, Slice regular functions on real alternative algebras, Adv. Math., 226 (2011), 1662–1691.
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Caterina Stoppato, Poles of regular quaternionic functions, Complex Var. Elliptic Equ. 54 (2009), no. 11, 1001–1018. MR 2572530, DOI 10.1080/17476930903275938
Bibliographic Information
- Fabrizio Colombo
- Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via Bonardi, 9, 20133 Milano, Italy
- MR Author ID: 601509
- Email: fabrizio.colombo@polimi.it
- Irene Sabadini
- Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via Bonardi, 9, 20133 Milano, Italy
- MR Author ID: 361222
- Email: irene.sabadini@polimi.it
- Daniele C. Struppa
- Affiliation: Department of Mathematics, Schmid College of Science, Chapman University, Orange, California 92866
- MR Author ID: 168380
- ORCID: 0000-0002-3664-1729
- Email: struppa@chapman.edu
- Received by editor(s): May 25, 2010
- Published electronically: December 13, 2010
- Communicated by: Franc Forstneric
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1787-1803
- MSC (2010): Primary 30G35; Secondary 30B10, 30C10
- DOI: https://doi.org/10.1090/S0002-9939-2010-10812-2
- MathSciNet review: 2763766