Mathematische Annalen

, Volume 363, Issue 3–4, pp 733–752 | Cite as

The Radon transform between monogenic and generalized slice monogenic functions

  • F. Colombo
  • R. Lávička
  • I. Sabadini
  • V. Souček


In Bureš et al. (Elements of quaternionic analysis and Radon transform, 2009), the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on \({\mathbb {R}}^{n+1}\) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra \({\mathbb {R}}_n\) is mapping solutions of the generalized Cauchy–Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in \({\mathbb {R}}_n\) and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.

Mathematics Subject Classification




The three co-authors (FC, IS and VS) thank the E. Čech Institute (the Grant P201/12/G028 of the Grant Agency of the Czech Republic) for the support during the preparation of the paper.


  1. 1.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Research Notes in Mathematics, vol. 76. Pitman, London (1982)Google Scholar
  2. 2.
    Bureš, J., Lávička, R., Souček, V.: Elements of Quaternionic Analysis and Radon Transform. Textos de Matematica 42, Departamento de Matematica, Universidade de Coimbra, Coimbra (2009)zbMATHGoogle Scholar
  3. 3.
    Colombo, F., Sabadini, I., Sommen, F.: The inverse Fueter mapping theorem. Commun. Pure Appl. Anal. 10, 1165–1181 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Colombo, F., Sabadini, I., Sommen, F.: The inverse Fueter mapping theorem in integral form using spherical monogenics. Israel J. Math. 194, 485–505 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Colombo, F., Sabadini, I., Struppa, D.C.: Slice monogenic functions. Israel J. Math. 171, 385–403 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics. 289, Birkhäuser/Springer Basel AG, Basel (2011)zbMATHCrossRefGoogle Scholar
  7. 7.
    Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions. Mathematics and Its Applications. Kluwer, Boston (1992)zbMATHCrossRefGoogle Scholar
  8. 8.
    Eelbode, D., Sommen, F.: The inverse Radon transform and the fundamental solution of the hyperbolic Dirac equation. Math. Zeit. 247, 733–745 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fueter, R.: Die Funktionentheorie der Differentialgleichungen \(\Delta u = 0\) und \(\Delta \Delta u = 0\) mit vier reellen Variablen. Comment. Math. Helv. 7, 307–330 (1934)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gelfand, I.M., Graev, M.I., Vilenkin, N.Y.: Generalized Functions, vol. 5. Integral Geometry and Representation Theory. Academic press, Boston (1966)Google Scholar
  11. 11.
    Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternion variable. Adv. Math. 216, 279–301 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)zbMATHCrossRefGoogle Scholar
  13. 13.
    Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic Functions in the Plane and \(n\)-Dimensional Space. Birkhäuser, Basel (2008)Google Scholar
  14. 14.
    Helgason, S.: Integral Geometry and Radon Transforms. Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
  15. 15.
    Morimoto, M.: Analytic functionals on the sphere and their Fourier-Borel transformations. Banach Cent. Publ. 11(1), 223–250 (1983)MathSciNetGoogle Scholar
  16. 16.
    Morimoto, M.: Analytic Functionals on the Sphere. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  17. 17.
    Seeley, R.T.: Spherical harmonics. Am. Math. Mon. 73(4), 115–121 (1966)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Sommen, F.: Spherical monogenic functions and analytic functionals on the unit sphere. Tokyo J. Math. 4(2), 427–456 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Sommen, F.: Plane elliptic monogenic functions in symmetric domains. Rend. Circ. Mat. Palermo 2, 259–269 (1984)MathSciNetGoogle Scholar
  20. 20.
    Sommen, F.: Plane wave decomposition of monogenic functions. Ann. Polon. Math. XLIX, 101–114 (1988)Google Scholar
  21. 21.
    Sommen, F.: An extension of the Radon transform to Clifford analysis. Complex Var. 8, 243–266 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Sommen, F.: Power series expansions of monogenic functions. Complex Var. 11, 215–222 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Sommen, F.: Radon and X-ray transforms in Clifford analysis. Complex Var. 11, 49–70 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Sommen, F.: Clifford analysis and integral geometry. In: Micali, A. et al.: Clifford Algebras and Their Applications in Mathematical Physics, pp. 293–311. Kluwer, Boston (1992)Google Scholar
  25. 25.
    Vilenkin, N.Y., Klimyk, A.U.: Representation of Lie Groups and Special Functions, vol. 2. Kluwer, Dordrecht (1993)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • F. Colombo
    • 1
  • R. Lávička
    • 2
  • I. Sabadini
    • 1
  • V. Souček
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

Personalised recommendations