Abstract
The purpose of this chapter is to describe the settings in which one can set up coherent generalizations to several variables of the classical single variable complex analysis and to point out some basic properties of the function spaces that naturally occur in such settings. Particular emphasis is placed on defining proper counterparts in higher dimension of the concept of holomorphic functions and of the Cauchy–Riemann operator. The resulting function theory deals with monogenic functions and Euclidean Dirac or Cauchy–Riemann operators. Its richness is illustrated by the existence of a general Cauchy–Pompeiu Integral Representation Formula. Actually, even more general formulas will be initially derived for first order homogeneous differential operators with constant coefficients in a unital associative Banach algebra. The point that will be made is that the operators and algebras that yield the simplest forms of such Integral Representation Formulas with Remainders are essentially the Cauchy–Riemann and Dirac operators associated with Euclidean Clifford algebras. As yet another generalization of a classical result in single variable complex analysis one gets a Quantitative Hartogs–Rosenthal Theorem concerned with uniform approximation on compact sets by monogenic functions. An equally important goal of the chapter is to outline some of the contributions that led to the discovery and development of Quaternion and Clifford analysis. The last section includes several concluding remarks and a few additional references that outline the full scope of some of the early and current developments, and help in further explorations of other lines of research in quaternion and Clifford analysis.
Similar content being viewed by others
References
Ablamowicz, R. (ed.): Clifford Algebras Applications to Mathematics, Physics, and Engineering. Progress in Mathematical Physics, vol. 34. Birkhäuser, Boston (2004)
Ablamowicz, R., Fauser, B. (eds.): Clifford Algebras and their Applications in Mathematical Physics, Volume 1: Algebra and Physics. Progress in Mathematical Physics, vol. 18. Birkhäuser, Boston (2000)
Ahlfors, L., Beurling, A.: Conformal invariants and function theoretic null sets. Acta Math.83, 101–129 ( 1950)
Alexander, H.: Projections of polynomial hulls. J. Funct. Anal.13, 13–19 (1973)
Alexander, H.: On the area of the spectrum of an element of a uniform algebra. In: Complex Approximation Proceedings, pp. 3–12, Quebec, 3–8 July 1978. Birkhäuser, Basel (1980)
Anglès, P.: Conformal Groups in Geometry and Spin Structures. Progress in Mathematical Physics, vol. 50. Birkhäuser, Boston (2008)
Bernstein, S.: A Borel-Pompeiu formula in \(\mathbb{C}^{n}\) and its applications to inverse scattering theory. In: Clifford Algebras and Their Applications in Mathematical Physics. Progress in Mathematical Physics Series, vol. 19, pp. 117–185. Birkhäuser, Boston (2000)
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Research Notes in Mathematics Series, vol. 76. Pitman, Massachusetts (1982)
Browder, F.: Functional analysis and partial differential equations. II. Math. Ann.145, 81–226 (1961)
Browder, F.: Approximation by solutions of partial differential equations. Am. J. Math.84, 134–160 (1962)
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)
Deavours, C.A.: The quaternion calculus. Am. Math. Mon.80, 995–1008 (1973)
Delanghe, R.: On regular-analytic functions with values in a Clifford algebra. Math. Ann.185, 91–111 (1970)
Delanghe, R.: On regular points and Liouville’s theorem functions with values in a Clifford algebra. Simon Stevin 44, 55–66 (1970–1971)
Delanghe, R.: On the singularities of functions with values in a Clifford algebra. Math. Ann.196, 293–319 (1972)
Delanghe, R., Krausshar, R.S., Malonek, H.R.: Differentiability of functions with values in some real associative algebras: Approaches to an old problem. Bulletin de la Société Royale des Sciences de Liège 70(4–6), 231–249 (2001)
Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer, Dordrecht (1982)
Fueter, R.: Analytische Funktionen einer Quaternionenvariablen. Comment. Math. Helv.4 , 9–20 (1932)
Fueter, R.: Die Funktionentheorie der Differentialgleichungen \(\Delta u = 0\) und \(\Delta \Delta u = 0\) mit vier reellen Variablen. Comment. Math. Helv.7, 307–330 (1935)
Fueter, R.: Über die analytische Darstellung der regulären Funktionen einer Quaternionen variablen. Comment. Math. Helv.8, 371–378 (1936)
Fueter, R.: Die Singularitäten der eindeutigen regulären Funktionen einer Quaternionen-variablen. Comment. Math. Helv.9, 320–335 (1937)
Gamelin, T.W.: Uniform Algebras. Prentice Hall, Englewood Cliffs (1969)
Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University Press, Cambridge (1991)
Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)
Haefeli, H.: Hyperkomplexe differentiale. Comment. Math. Helv.20, 382–420 (1947)
Hamilton, W.R.: Elements of Quaternions. Longmans Green, London (1866)
Hartogs, F., Rosenthal, A.: Über Folgen analytischer Functionen. Math. Ann.104, 606–610 (1931)
Hausdorff, F.: Zür Theorie der Systeme complexer Zahlen. Leipz. Ber.52, 43–61 (1900)
Hile, G.N.: Representations of solutions of a special class of first order systems. J. Differ. Equ.25, 410–424 (1977)
Iftimie, V.: Fonctions hypercomplexes. Bull. Math. Soc. Sci. Math RSR 57, 279–332 (1965)
Krausshar, R.S.: Generalized Analytic Automorphic Forms in Hypercomplex Spaces. Frontiers in Mathematics Series, vol. 15. Birkhäuser, Basel (2004)
Kravchenko, V.V., Shapiro, M.V.: Integral Representations for Spatial Models of Mathematical Physics. Pitman Research Notes in Mathematics Series, vol. 351. Longman, Harlow (1996)
Krylov, N.M.: On Rowan Hamilton’s quaternions and the notion of monogenicity (in Russian). Dokl. Akad. Nauk SSSR 55, 799–800 (1947)
Malonek, H.R.: Representations of solutions of a special class of first order systemsA new hypercomplex structure of the Euclidean space \(\mathbb{R}^{m+1}\) and the concept of hypercomplex differentiability. Complex Var.14, 25–33 (1990)
Malonek, H.R.: Power series representations for monogenic functions in \(\mathbb{R}^{m+1}\) based on a permutation product. Complex Var.15, 181–191 (1990)
Martin, M.: Higher-dimensional Ahlfors-Beurling inequalities in Clifford analysis. Proc. Am. Math. Soc.126, 2863–2871 (1998)
Martin, M.: Convolution and maximal operator inequalities. In: Clifford Algebras and Their Applications in Mathematical Physics. Progress in Mathematical Physics, vol. 19, pp. 83–100. Birkhäuser, Boston (2000)
Martin, M.: Spin geometry, Clifford analysis, and joint seminormality. In: Advances in Analysis and Geeometry. Trends in Mathematics Series, vol. 1, pp. 227–255. Birkhäuser, Boston (2004)
Martin, M.: Uniform approximation by solutions of elliptic equations and seminormality in higher dimensions. Operator Theory: Advances and Applications, vol. 149, pp. 387–406. Birkhäuser, Basel (2004)
Martin, M.: Uniform approximation by closed forms in several complex variables. Adv. Appl. Clifford Algebras 19 (3–4), 777–792 (2009)
Martin, M.: Deconstructing Dirac operators. I: Quantitative Hartogs-Rosenthal theorems. In: More Progress in Analysis. Proceedings of the Fifth International Society for Analysis, Its Applications and Computation Congress, ISAAC 2005, pp. 1065–1074. World Scientific, Singapore (2009)
Martin, M.: Deconstructing Dirac operators. III: Dirac and semi-Dirac pairs. In: Operator Theory: Advances and Applications, vol. 203, pp. 347–362. Birkhäuser, Basel (2009)
Martin, M.: Deconstructing Dirac operators. II: Integral representation formulas. In: Hypercomplex Analysis and Applications. Trends in Mathematics; Proceedings of the 7th International Society for Analysis, Its Applications and Computation Congress, ISAAC 2009, London, pp. 195–211. Springer, Basel (2011)
Mejlihzon, A.Z.: On the notion of monogenic quaternions (in Russian). Dokl. Akad. Nauk SSSR 59, 431–434 (1948)
Mitrea, M.: Singular Integrals, Hardy Spaces, and Clifford Wavelets. Lecture Notes in Mathematics, vol. 1575. Springer, Berlin (1994)
Qian, T., Hempfling, Th., McIntosh, A., Sommen, F. (eds.): Advances in Analysis and Geometry. New Developments Using Clifford Algebras. Trends in Mathematics Series, vol. 14. Birkhäuser, Basel (2004)
Rocha-Chavez, R., Shapiro M., Sommen, F.: On the singular Bochner-Martinelli integral. Integr. Equ. Oper. Theory 32, 354–365 (1998)
Rocha-Chavez, R., Shapiro M., Sommen, F.: Analysis of functions and differential forms in \(\mathbb{C}^{m}\). In: Proceedings of the Second International Society for Analysis, Its Applications and Computation Congress, ISAAC 1999, pp. 1457–1506. Kluwer, Dordrecht (2000)
Rocha-Chavez, R., Shapiro M., Sommen, F.: Integral theorems for solutions of the complex Hodge-Dolbeault system. In: Proceedings of the Second International Society for Analysis, Its Applications and Computation Congress, ISAAC 1999, pp. 1507–1514. Kluwer, Dordrecht (2000)
Rocha-Chavez, R., Shapiro M., Sommen, F.: Integral Theorems for Functions and Differential Forms in \(\mathbb{C}^{m}\). Research Notes in Mathematics, vol. 428. Chapman & Hall, Boca Raton (2002)
Ryan, J.: Applications of complex Clifford analysis to the study of solutions to generalized Dirac and Klein-Gordon equations, with holomorphic potential. J. Differ. Equ.67, 295–329 (1987)
Ryan, J.: Cells of harmonicity and generalized Cauchy integral formulae. Proc. Lond. Math. Soc.60, 295–318 (1990)
Ryan, J.: Plemelj formulae and transformations associated to plane wave decompositions in complex Clifford analysis. Proc. Lond. Math. Soc.64, 70–94 (1991)
Ryan, J. (ed.): Clifford Algebras in Analysis and Related Topics. CRC Press, Boca Raton (1995)
Ryan, J.: Intrinsic Dirac operators in \(\mathbb{C}^{n}\). Adv. Math.118, 99–133 (1996)
Ryan, J., Sprössig, W. (eds.): Clifford Algebras and Their Applications in Mathematical Physics, Volume 2: Clifford Analysis. Progress in Mathematical Physics, vol. 19. Birkhäuser, Boston (2000)
Sabadini, I., Shapiro, M., Sommen, F. (eds.): Hypercomplex Analysis. Trends in Mathematics Series, vol. 6. Birkhäuser, Basel (2009)
Scheffers, G.: Verallgemeinerung der Grundlagen der gewöhnlichen complexen Zahlen. Berichte kgl. Sächs. Ges. der Wiss.52, 60 (1893)
Shapiro, M.: Some remarks on generalizations of the one-dimensional complex analysis: hypercomplex approach. In: Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations, pp. 379–401. World Scientific, Singapore (1995)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)
Sommen, F.: Martinelli-Bochner formulae in complex Clifford analysis. Zeitschrift für Analysis und ihre Anwendungen 6, 75–82 (1987)
Sommen, F.: Defining a q-deformed version of Clifford analysis. Complex Var. Theory Appl.34, 247–265 (1997)
Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc.85, 199–225 (1979)
Tarkhanov, N.N.: The Cauchy Problem for Solutions of Elliptic Equations. Akademie, Berlin (1995)
Vasilevski, N., Shapiro, M.: Some questions of hypercomplex analysis. In: Complex Analysis and Applications, Sofia, 1987, pp. 523–531 (1989)
Weinstock, B.M.: Uniform approximations by solutions of elliptic equations. Proc. Am. Math. Soc.41, 513–517 (1973)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Basel
About this entry
Cite this entry
Martin, M. (2014). Function Spaces in Quaternionic and Clifford Analysis. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_24-1
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0692-3_24-1
Received:
Accepted:
Published:
Publisher Name: Springer, Basel
Online ISBN: 978-3-0348-0692-3
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering