Generalized Partial-Slice Monogenic Functions: A Synthesis of Two Function Theories
In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698 , 2023). The class of these functions includes both the theory of monogenic functions and of sli...
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| Published in: | Advances in applied Clifford algebras Vol. 34; no. 2; p. 10 |
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| Abstract | In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions,
arXiv:2309.03698
, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of
∗
-product based on the CK-extension and discuss the smoothness of generalized partial-slice functions. |
|---|---|
| AbstractList | In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions,
arXiv:2309.03698
, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of
$$*$$
∗
-product based on the CK-extension and discuss the smoothness of generalized partial-slice functions. In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698 , 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of ∗ -product based on the CK-extension and discuss the smoothness of generalized partial-slice functions. In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of ∗-product based on the CK-extension and discuss the smoothness of generalized partial-slice functions. |
| ArticleNumber | 10 |
| Author | Xu, Zhenghua Sabadini, Irene |
| Author_xml | – sequence: 1 givenname: Zhenghua surname: Xu fullname: Xu, Zhenghua organization: School of Mathematics, Hefei University of Technology – sequence: 2 givenname: Irene orcidid: 0000-0002-9930-4308 surname: Sabadini fullname: Sabadini, Irene email: irene.sabadini@polimi.it organization: Dipartimento di Matematica, Politecnico di Milano |
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| Keywords | Primary 30G35 Secondary 32A30 Slice monogenic functions Clifford algebras Functions of a hypercomplex variable Monogenic functions |
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| References | Colombo, F., Sabadini, I., Struppa, D.C.: Michele Sce’s Works in Hypercomplex Analysis. A Translation with Commentaries. Birkhäuser, Basel (2020) Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, vol. 39. Birkhäuser, Boston (2004) Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions, Progress in Mathematics, vol. 289. Birkhäuser/Springer, Basel (2011) ColomboFSabadiniIStruppaDCSlice monogenic functionsIsr. J. Math.2009171385403252011610.1007/s11856-009-0055-4 BrackxFDelangheRSommenFClifford analysis, Research Notes in Mathematics1982BostonPitman WhitneyHDifferentiable even functionsDuke Math. J.194310159160778310.1215/S0012-7094-43-01015-4 Delanghe, R., Sommen, F., Souček, V.: Clifford algebra and spinor-valued functions. A function theory for the Dirac operator, Mathematics and its Applications, vol. 53. Kluwer Academic Publishers Group, Dordrecht (1992) GentiliG StruppaDCA new approach to Cullen-regular functions of a quaternionic variableC. R. Math. Acad. Sci. Paris200634210741744222775110.1016/j.crma.2006.03.015 SceMOsservazioni sulle serie di potenze nei moduli quadraticiAtti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat.19572322022597386 Xu, Z., Sabadini, I.: On the Fueter–Sce theorem for generalized partial-slice monogenic functions. arXiv:2311.12545 (2023) FueterRDie funktionentheorie der differentialgleichungen Δu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta u = 0$$\end{document} und ΔΔu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \Delta u = 0$$\end{document} mit vier reellen variablenComment. Math. Helv.19347307330150951510.1007/BF01292723 Xu, Z., Sabadini, I.: Generalized partial-slice monogenic functions. arXiv:2309.03698 (2023) GürlebeckKHabethaKSprößigWHolomorphic Functions in the Plane and n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-dimensional Space2008BaselBirkhäuser Verlag GhiloniRPerottiASlice regular functions on real alternative algebrasAdv. Math.2011226216621691273779610.1016/j.aim.2010.08.015 Moisil, Gr.C.: Sur les quaternions monogènes. Bull. Sci. Math. 55, 168–174 (1931) F Colombo (1314_CR3) 2009; 171 R Ghiloni (1314_CR9) 2011; 226 1314_CR6 1314_CR14 1314_CR5 1314_CR15 1314_CR2 R Fueter (1314_CR7) 1934; 7 1314_CR11 1314_CR4 M Sce (1314_CR12) 1957; 23 F Brackx (1314_CR1) 1982 G Gentili (1314_CR8) 2006; 342 K Gürlebeck (1314_CR10) 2008 H Whitney (1314_CR13) 1943; 10 |
| References_xml | – reference: ColomboFSabadiniIStruppaDCSlice monogenic functionsIsr. J. Math.2009171385403252011610.1007/s11856-009-0055-4 – reference: BrackxFDelangheRSommenFClifford analysis, Research Notes in Mathematics1982BostonPitman – reference: GürlebeckKHabethaKSprößigWHolomorphic Functions in the Plane and n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-dimensional Space2008BaselBirkhäuser Verlag – reference: Colombo, F., Sabadini, I., Struppa, D.C.: Michele Sce’s Works in Hypercomplex Analysis. A Translation with Commentaries. Birkhäuser, Basel (2020) – reference: GentiliG StruppaDCA new approach to Cullen-regular functions of a quaternionic variableC. R. Math. Acad. Sci. Paris200634210741744222775110.1016/j.crma.2006.03.015 – reference: Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions, Progress in Mathematics, vol. 289. Birkhäuser/Springer, Basel (2011) – reference: Delanghe, R., Sommen, F., Souček, V.: Clifford algebra and spinor-valued functions. A function theory for the Dirac operator, Mathematics and its Applications, vol. 53. Kluwer Academic Publishers Group, Dordrecht (1992) – reference: GhiloniRPerottiASlice regular functions on real alternative algebrasAdv. Math.2011226216621691273779610.1016/j.aim.2010.08.015 – reference: Xu, Z., Sabadini, I.: Generalized partial-slice monogenic functions. arXiv:2309.03698 (2023) – reference: WhitneyHDifferentiable even functionsDuke Math. J.194310159160778310.1215/S0012-7094-43-01015-4 – reference: Xu, Z., Sabadini, I.: On the Fueter–Sce theorem for generalized partial-slice monogenic functions. arXiv:2311.12545 (2023) – reference: Moisil, Gr.C.: Sur les quaternions monogènes. Bull. Sci. Math. 55, 168–174 (1931) – reference: SceMOsservazioni sulle serie di potenze nei moduli quadraticiAtti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat.19572322022597386 – reference: Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, vol. 39. Birkhäuser, Boston (2004) – reference: FueterRDie funktionentheorie der differentialgleichungen Δu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta u = 0$$\end{document} und ΔΔu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \Delta u = 0$$\end{document} mit vier reellen variablenComment. Math. Helv.19347307330150951510.1007/BF01292723 – ident: 1314_CR2 doi: 10.1007/978-0-8176-8166-1 – ident: 1314_CR4 doi: 10.1007/978-3-0348-0110-2 – ident: 1314_CR15 doi: 10.1007/s10231-024-01508-1 – volume: 23 start-page: 220 year: 1957 ident: 1314_CR12 publication-title: Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. – volume: 171 start-page: 385 year: 2009 ident: 1314_CR3 publication-title: Isr. J. Math. doi: 10.1007/s11856-009-0055-4 – ident: 1314_CR6 doi: 10.1007/978-94-011-2922-0 – volume-title: Clifford analysis, Research Notes in Mathematics year: 1982 ident: 1314_CR1 – volume: 7 start-page: 307 year: 1934 ident: 1314_CR7 publication-title: Comment. Math. Helv. doi: 10.1007/BF01292723 – volume-title: Holomorphic Functions in the Plane and $$n$$-dimensional Space year: 2008 ident: 1314_CR10 – volume: 10 start-page: 159 year: 1943 ident: 1314_CR13 publication-title: Duke Math. J. doi: 10.1215/S0012-7094-43-01015-4 – ident: 1314_CR5 doi: 10.1007/978-3-030-50216-4 – ident: 1314_CR14 – volume: 342 start-page: 741 issue: 10 year: 2006 ident: 1314_CR8 publication-title: C. R. Math. Acad. Sci. Paris doi: 10.1016/j.crma.2006.03.015 – volume: 226 start-page: 1662 issue: 2 year: 2011 ident: 1314_CR9 publication-title: Adv. Math. doi: 10.1016/j.aim.2010.08.015 – ident: 1314_CR11 |
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| SubjectTerms | Algebra Applications of Mathematics Euclidean space Mathematical and Computational Physics Mathematical Methods in Physics Operators (mathematics) Physics Physics and Astronomy Smoothness Synthesis Theoretical |
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