Journal of the Mathematical Society of Japan

A Fourier–Borel transform for monogenic functionals

Irene SABADINI and Franciscus SOMMEN

Full-text: Open access


We discuss the Fourier–Borel transform for the dual of spaces of monogenic functions. This transform may be seen as a restriction of the classical Fourier–Borel transform for holomorphic functionals, and it transforms spaces of monogenic functionals into quotients of spaces of entire holomorphic functions of exponential type. We prove that, for the Lie ball, these quotient spaces are isomorphic to spaces of monogenic functions of exponential type.

Article information

J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1487-1504.

First available in Project Euclid: 24 October 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30G35: Functions of hypercomplex variables and generalized variables 46F15: Hyperfunctions, analytic functionals [See also 32A25, 32A45, 32C35, 58J15] 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Fourier–Borel transform Lie ball analytic functionals monogenic functionals


SABADINI, Irene; SOMMEN, Franciscus. A Fourier–Borel transform for monogenic functionals. J. Math. Soc. Japan 68 (2016), no. 4, 1487--1504. doi:10.2969/jmsj/06841487.

Export citation


  • J. Aniansson, Some integral representations in real and complex analysis. Peano–Sard kernels and Fischer kernels, Doctoral thesis, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden, 1999.
  • R. Delanghe and F. Brackx, Duality in hypercomplex function theory, J. Funct. Anal., 37 (1980), 164–181.
  • F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Res. Notes in Math., 76, Pitman (Advanced Publishing Program), Boston, MA, 1982.
  • F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, Analysis of Dirac Systems and Computational Algebra, Progress in Math. Physics, 39, Birkhäuser, Boston, 2004.
  • F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, Twisted plane wave expansions using hypercomplex methods, Publ. RIMS Kyoto Univ., 50 (2014), 1–18.
  • N. De Schepper and F. Sommen, Closed form of the Fourier–Borel kernel in the framework of Clifford analysis, Results Math., 62 (2012), 181–202.
  • K. I. Kou and T. Qian, The Paley–Wiener theorem in ${\bold R}^n$ with the Clifford analysis setting, J. Funct. Anal., 189 (2002), 227–241.
  • C. Li, A. McIntosh and T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana, 10 (1994), 665–721.
  • A. Martineau, Sur les fonctionelles analytiques et la transformation de Fourier–Borel, J. Anal. Math., 11 (1963), 1–164.
  • M. Morimoto, Analytic functionals on the Lie sphere, Tokyo J. Math., 3 (1980), 1–35.
  • I. Sabadini, F. Sommen and D. C. Struppa, Sato's hyperfunctions and boundary values of monogenic functions, Adv. Appl. Clifford Alg., 24 (2014), 1131–1143.
  • H. S. Shapiro, An algebraic theorem of G. Fischer, and the holomorphic Goursat problem, Bull. London Math. Soc., 21 (1989), 513–537.
  • J. Siciak, Holomorphic continuation of harmonic functions, Ann. Polon. Math., 29 (1974), 67–73.
  • F. Sommen, A product and an exponential function in hypercomplex function theory, Appl. Anal., 12 (1981), 13–26.
  • F. Sommen, Hyperfunctions with values in a Clifford algebra, Simon Stevin, Quart. J. Pure Appl. Math., 57 (1983), 225–254.
  • F. Sommen, Microfunctions with values in a Clifford algebra, II, Sci. Papers College of Arts and Sciences, Univ. Tokyo, 36 (1986), 15–37.
  • F. Sommen, Spherical monogenic on the Lie sphere, J. Funct. Anal., 92 (1990), 372–402.