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The Hilbert Transform on the Unit Sphere in ℝm

  • F. Brackx
  • H. De Schepper
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

As an intrinsically multidimensional function theory, Clifford analysis offers a framework which is particularly suited for the integrated treatment of higher-dimensional phenomena. In this paper a detailed account is given of results connected to the Hilbert transform on the unit sphere in Euclidean space and some of its related concepts, such as Hardy spaces and the Cauchy integral, in a Clifford analysis context.

Keywords

Hilbert transform Hardy space Cauchy integral 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 44A15 

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References

  1. [1]
    S. Bernstein, L. Lanzani, Szegö projections for Hardy spaces of monogenic functions and applications, IJMMS 29(10), 2002, 613–624.zbMATHMathSciNetGoogle Scholar
  2. [2]
    S.R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping, CRC Press (Boca Raton-Ann Arbor-London-Tokyo, 1992).Google Scholar
  3. [3]
    F. Brackx, B. De Knock, H. De Schepper, D. Eelbode, On the interplay between the Hilbert transform and conjugate harmonic functions, Math. Meth. Appl. Sci. 29(12), 2006, 1435–1450.zbMATHCrossRefGoogle Scholar
  4. [4]
    F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Pitman Advanced Publishing Program (Boston-London-Melbourne, 1982).Google Scholar
  5. [5]
    F. Brackx, H. De Schepper, The Hilbert Transform on a Smooth Closed Hypersurface, CUBO, A Mathematical Journal 10(2), 2008, 83–106.zbMATHMathSciNetGoogle Scholar
  6. [6]
    F. Brackx, H. De Schepper, D. Eelbode, A new Hilbert transform on the unit sphere in ℝm, Complex Var. Elliptic Equ. 51(5-6), 2006, 453–462.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    F. Brackx, N. Van Acker, H p spaces of monogenic functions. In: A. Micali et al. (eds.), Clifford Algebras and their Applications in Mathematical Physics, Kluwer Academic Publishers (Dordrecht, 1992), 177–188.Google Scholar
  8. [8]
    P. Calderbank, Clifford analysis for Dirac operators on manifolds-with-boundary, Max Planck-Institut für Mathematik (Bonn, 1996).Google Scholar
  9. [9]
    J. Cnops, An introduction to Dirac operators on manifolds, Birkhäuser Verlag (Basel, 2002).Google Scholar
  10. [10]
    R. Delanghe, Some remarks on the principal value kernel in ℝm, Complex Var. Theory Appl. 47, 2002, 653–662.zbMATHMathSciNetGoogle Scholar
  11. [11]
    R. Delanghe, On the Hardy spaces of harmonic and monogenic functions in the unit ball of ℝm+1. In: Acoustics, mechanics and the related topics of mathematical analysis, World Scientific Publishing (River Edge, New Jersey, 2002), 137–142.Google Scholar
  12. [12]
    R. Delanghe, On Some Properties of the Hilbert Transform in Euclidean Space, Bull. Belg. Math. Soc.-Simon Stevin 11, 2004, pp. 163–180.zbMATHMathSciNetGoogle Scholar
  13. [13]
    R. Delanghe, F. Sommen, V. Souček, Clifford Algebra and Spinor-Valued Functions, Kluwer Academic Publishers (Dordrecht-Boston-London, 1992).Google Scholar
  14. [14]
    J. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press (Cambridge, 1991).Google Scholar
  15. [15]
    K. Gürlebeck, W. Sprößig, Quaternionic analysis and elliptic boundary value problems, Birkhäuser Verlag (Basel, 1990).Google Scholar
  16. [16]
    K. Gürlebeck, W. Sprößig, Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley (Chichester, 1998).Google Scholar
  17. [17]
    K. Gürlebeck, K. Habetha and W. Sprößig, Funktionentheorie in der Ebene und im Raum, Birkhäuser Verlag (Basel, 2006).Google Scholar
  18. [18]
    S.L. Hahn, Hilbert Transforms in Signal Processing, Artech House (Boston-London, 1996).Google Scholar
  19. [19]
    S. Hofmann, E. Marmolejo Olea, M. Mitrea, S. Pérez Esteva, M. Taylor, Hardy Spaces, Singular Integrals and the Geometry of Euclidean Domains of Locally Finite Perimeter, to appear.Google Scholar
  20. [20]
    V.V. Kravchenko, M.V. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Pitman Research Notes in Mathematics Series 351, Longman Scientific and Technical (Harlow, 1996).Google Scholar
  21. [21]
    C. Li, A. McIntosh, T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Math. Iberoamer. 10, 1994, 665–721.zbMATHMathSciNetGoogle Scholar
  22. [22]
    C. Li, A. McIntosh, S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5, 1992, 455–481.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    A. McIntosh, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains. In: J. Ryan (ed.), Clifford Algebras in Analysis and Related Topics, Studies in Advanced Mathematics, CRC Press (Boca Raton, 1996), 33–87.Google Scholar
  24. [24]
    S.G. Mikhlin, Mathematical Physics, an Advanced course, North-Holland Publ. Co. (Amsterdam-London, 1970).Google Scholar
  25. [25]
    M. Mitrea, Clifford Wavelets, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics 1575, Springer-Verlag (Berlin, 1994).Google Scholar
  26. [26]
    M. Murray, The Cauchy integral, Calderon commutation, and conjugation of singular integrals in ℝn, Trans. of the AMS 298, 1985, 497–518.CrossRefGoogle Scholar
  27. [27]
    T. Qian et al. (eds.), Advances in analysis and geometry: new developments using Clifford algebras, Birkhäuser Verlag (Basel-Boston-Berlin, 2004).Google Scholar
  28. [28]
    J. Ryan (ed.), Clifford Algebras in Analysis and Related Topics, Studies in Advanced Mathematics, CRC Press (Boca Raton, 1996).Google Scholar
  29. [29]
    J. Ryan, Basic Clifford Analysis, CUBO, A Mathematical Journal 2, 2000, 226–256.zbMATHGoogle Scholar
  30. [30]
    J. Ryan, Clifford Analysis. In: R. Ablamowicz and G. Sobczyk (eds.), Lectures on Clifford (Geometric) Algebras and Applications, Birkhäuser (Boston-Basel-Berlin, 2004), 53–89.Google Scholar
  31. [31]
    J. Ryan, D. Struppa (eds.), Dirac operators in analysis, Addison Wesley Longman Ltd. (Harlow, 1998).Google Scholar
  32. [32]
    M.V. Shapiro, N.L. Vasilevski, Quaternionic ψ-holomorphic functions, singular integral operators and boundary value problems, Parts I and II, Complex Var. Theory Appl. 27, 1995, 14–46 and 67–96.Google Scholar
  33. [33]
    E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press (Princeton, 1993).Google Scholar
  34. [34]
    P. Van Lancker, The Kerzman-Stein Theorem on the Sphere, Complex Var. Theory and Appl. 45(1), 2001, 73–99.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • F. Brackx
    • 1
  • H. De Schepper
    • 1
  1. 1.Clifford Research Group Faculty of EngineeringGhent UniversityGentBelgium

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