Abstract
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.
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References
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Research Notes in Mathematics, vol. 76. Pitman, London (1982)
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)
Colombo, F., Sabadini, I., Sommen, F.: The inverse Fueter mapping theorem. Commun. Pure Appl. Anal. 10, 1165–1181 (2011)
Colombo, F., Sabadini, I., Sommen, F.: The inverse Fueter mapping theorem in integral form using spherical monogenics. Israel J. Math. 194, 485–505 (2013)
Colombo, F., Sabadini, I., Struppa, D.C.: Slice monogenic functions. Israel J. Math. 171, 385–403 (2009)
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)
Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions. Mathematics and Its Applications. Kluwer, Boston (1992)
Eelbode, D., Sommen, F.: The inverse Radon transform and the fundamental solution of the hyperbolic Dirac equation. Math. Zeit. 247, 733–745 (2004)
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)
Gelfand, I.M., Graev, M.I., Vilenkin, N.Y.: Generalized Functions, vol. 5. Integral Geometry and Representation Theory. Academic press, Boston (1966)
Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternion variable. Adv. Math. 216, 279–301 (2007)
Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic Functions in the Plane and \(n\)-Dimensional Space. Birkhäuser, Basel (2008)
Helgason, S.: Integral Geometry and Radon Transforms. Springer, Berlin (2011)
Morimoto, M.: Analytic functionals on the sphere and their Fourier-Borel transformations. Banach Cent. Publ. 11(1), 223–250 (1983)
Morimoto, M.: Analytic Functionals on the Sphere. American Mathematical Society, Providence (1998)
Seeley, R.T.: Spherical harmonics. Am. Math. Mon. 73(4), 115–121 (1966)
Sommen, F.: Spherical monogenic functions and analytic functionals on the unit sphere. Tokyo J. Math. 4(2), 427–456 (1981)
Sommen, F.: Plane elliptic monogenic functions in symmetric domains. Rend. Circ. Mat. Palermo 2, 259–269 (1984)
Sommen, F.: Plane wave decomposition of monogenic functions. Ann. Polon. Math. XLIX, 101–114 (1988)
Sommen, F.: An extension of the Radon transform to Clifford analysis. Complex Var. 8, 243–266 (1987)
Sommen, F.: Power series expansions of monogenic functions. Complex Var. 11, 215–222 (1989)
Sommen, F.: Radon and X-ray transforms in Clifford analysis. Complex Var. 11, 49–70 (1989)
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)
Vilenkin, N.Y., Klimyk, A.U.: Representation of Lie Groups and Special Functions, vol. 2. Kluwer, Dordrecht (1993)
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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.
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Colombo, F., Lávička, R., Sabadini, I. et al. The Radon transform between monogenic and generalized slice monogenic functions. Math. Ann. 363, 733–752 (2015). https://doi.org/10.1007/s00208-015-1182-3
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DOI: https://doi.org/10.1007/s00208-015-1182-3