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Complex Analysis and Operator Theory

, Volume 6, Issue 5, pp 1047–1067 | Cite as

The Fractional Clifford-Fourier Transform

  • Hendrik De BieEmail author
  • Nele De Schepper
Article

Abstract

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.

Keywords

Clifford-Fourier transform Fractional transform Integral kernel Clifford analysis 

Mathematics Subject Classification

30G35 42B10 

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References

  1. 1.
    Bargmann V.: On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, 187–214 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ben Saïd, S., Kobayashi, T., Ørsted, B.: Laguerre semigroup and Dunkl operators. Compositio Math. arXiv:0907.3749 (to appear)Google Scholar
  3. 3.
    Brackx F., De Schepper N., Sommen F.: The Clifford-Fourier transform. J. Fourier Anal. Appl. 11, 669–681 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brackx F., De Schepper N., Sommen F.: The two-dimensional Clifford-Fourier transform. J. Math. Imaging Vis. 26, 5–18 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brackx F., De Schepper N., Sommen F.: The Fourier transform in Clifford analysis. Adv. Imaging Electron Phys. 156, 55–203 (2008)CrossRefGoogle Scholar
  6. 6.
    Colombo, F., De Bie, H.: The S-spectrum and the Clifford-Fourier transform (in preparation)Google Scholar
  7. 7.
    Condon E.U.: Immersion of the Fourier transform in a continuous group of functional transformations. Proc. Nat. Acad. Sci. USA 23, 158–164 (1937)CrossRefGoogle Scholar
  8. 8.
    De Bie, H., De Schepper, N.: Fractional Fourier transforms of hypercomplex signals. Signal Image Video Process 2012 (accepted for publication)Google Scholar
  9. 9.
    De Bie, H., De Schepper, N., Sommen, F.: The class of Clifford-Fourier transforms. J. Fourier Anal. Appl. 17, 1198–1231 (2011)Google Scholar
  10. 10.
    De Bie, H., Xu, Y.: On the Clifford-Fourier transform. Int. Math. Res. Not. IMRN 2011, 5123–5163 (2011)Google Scholar
  11. 11.
    Delanghe, R., Sommen, F., Souček, V.: Clifford algebra and spinor-valued functions. Mathematics and its applications, vol. 53. Kluwer Academic Publishers Group, DordrechtGoogle Scholar
  12. 12.
    Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Higher transcendental functions, vol. II. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  13. 13.
    Folland G.B.: Harmonic analysis in phase space. Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)Google Scholar
  14. 14.
    Gradshteyn I.S., Ryzhik I.M.: Table of integrals, series, and products. Academic Press, New York (1980)zbMATHGoogle Scholar
  15. 15.
    Howe, R.: The oscillator semigroup. In: The mathematical heritage of Hermann Weyl (Durham, NC, 1987). Proc. Sympos. Pure Math., vol. 48, pp. 61–132. Am. Math. Soc., Providence (1988)Google Scholar
  16. 16.
    Mehler F.G.: . J. Reine Angew. Math. 66, 161–176 (1866)zbMATHCrossRefGoogle Scholar
  17. 17.
    Mustard D.: Fractional convolution. J. Aust. Math. Soc. Ser. B 40, 257–265 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Namias V.: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 25(3), 241–265 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ozaktas H., Zalevsky Z., Kutay M.: The fractional Fourier transform. Wiley, Chichester (2001)Google Scholar
  20. 20.
    Sommen F.: Special functions in Clifford analysis and axial symmetry. J. Math. Anal. Appl. 130(1), 110–133 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Szegő G.: Orthogonal Polynomials, 4th edn. Am. Math. Soc. Colloq. Publ, Providence (1975)Google Scholar
  22. 22.
    Thangavelu, S.: Hermite and Laguerre semigroups: some recent developments. CIMPA-Venezuela lecture notes (to appear)Google Scholar
  23. 23.
    Walters S.: Periodic integral transforms and C *-algebras. C. R. Math. Acad. Sci. Soc. R. Can. 26, 55–61 (2004)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Watson G.N.: A treatise on the theory of Bessel Functions. Cambridge University Press/The Macmillan Company, Cambridge/New York (1944)zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mathematical Analysis, Faculty of Engineering and ArchitectureGhent UniversityGentBelgium

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