Spectrums of Functions Associated to the Fractional Clifford–Fourier Transform

  • Shanshan Li
  • Jinsong Leng
  • Minggang FeiEmail author


In this paper, we prove several versions of the real Paley–Wiener theorems for a fractional Clifford–Fourier transform which depends on two numerical parameters and paves the way in some sense for a functional calculus approach to generalizing the Fourier transform to Clifford analysis.


Clifford analysis Fractional Clifford–Fourier transform Real Paley–Wiener theorem 

Mathematics Subject Classification

42B10 30G30 



  1. 1.
    Andersen, N.B.: Real Paley–Wiener theorems. Bull. Lond. Math. Soc. 36, 504–508 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andersen, N.B.: Real Paley–Wiener theorems for the Dunkl transform on \({{\mathbb{R}}}\). Integral Transform Spec. Funct. 17, 543–547 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andersen, N.B., de Jeu, M.: Real Paley–Wiener theorems and local spectral radius formulas. Trans. Am. Math. Soc. 362, 3613–3640 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bang, H.H.: A property of infinitely differentiable functions. Proc. Am. Math. Soc. 108, 73–76 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics 76. Boston, MA: Pitman (Advanced Publishing Program) (1982)Google Scholar
  6. 6.
    Brackx, F., De Schepper, N., Sommen, F.: The Clifford–Fourier transform. J. Fourier Anal. Appl. 11(6), 669–681 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brackx, F., De Schepper, N., Sommen, F.: The two-dimensional Clifford–Fourier transform. J. Math. Imaging Vision 26(1–2), 5–18 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brackx, F., De Schepper, N., Sommen, F.: The Fourier transform in Clifford analysis. Adv. Imaging Electron Phys. 156, 55–203 (2008)CrossRefGoogle Scholar
  9. 9.
    Chen, Q.H., Li, L.Q., Ren, G.B.: Generalized Paley–Wiener theorems. Int. J. Wavelets, Multiresolut. Inf. Process. 10(2), 1250020 (2012) (7 pages)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chettaoui, C., Othmani, Y., Triméche, K.: On the range of the Dunkl transform on \({\mathbb{R}}\). Appl. Anal. (Singap.) 2(3), 177–192 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    De Bie, H., Xu, Y.: On the Clifford–Fourier transform. Int. Math. Res. Not. IMRN 22, 5123–5163 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    De Bie, H., De Schepper, N.: The fractional Clifford–Fourier transform. Complex Anal. Oper. Theory 6(5), 1047–1067 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    De Bie, H., De Schepper, N., Sommen, F.: The class of Clifford–Fourier transforms. J. Fourier Anal. Appl. 17(6), 1198–1231 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Delanghe, R., Sommen, F., Soucek, V.: Clifford algebra and spinor valued functions: a function theory for Dirac operator. Kluwer, Dordrecht (1992)CrossRefGoogle Scholar
  15. 15.
    Fei, M., Xu, Y., Yan, J.: Real Paley–Wiener theorem for the quaternion Fourier transform. Complex Var. Elliptic Equ. 62(8), 1072–1080 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fu, Y.X., Li, L.Q.: Paley–Wiener and Boas theorems for the quaternion Fourier transform. Adv. Appl. Clifford Algebr. 23(4), 837–848 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fu, Y.X., Li, L.Q.: Real Paley–Wiener theorems for the Clifford Fourier transform. Sci. China Math. 57(11), 2381–2392 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kou, K.-I., Qian, T.: The Paley–Wiener theorem in \({\mathbb{R}}^n\) with the Clifford analysis setting. J. Funct. Anal. 189(1), 227–241 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Li, S., Leng, J., Fei, M.: Paley–Wiener-type theorems for the Clifford-Fourier transform. Math. Methods Appl. Sci. 42(18), 6101–6113 (2019)CrossRefGoogle Scholar
  20. 20.
    Paley, R., Wiener, N.: The Fourier transforms in the complex domain, Amer. Math. Soc., Colloq. Publ. Ser., Vol. 19, Providence, RI (1934)Google Scholar
  21. 21.
    Qian, T.: Paley-Wiener theorems and Shannon sampling in Clifford analysis setting. Clifford algebras (Cookeville, TN, 2002), 115-124, Prog. Math. Phys., 34, Birkhser Boston, Boston, MA (2004)Google Scholar
  22. 22.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton (1971)zbMATHGoogle Scholar
  23. 23.
    Tuan, V.K.: On the Paley-Wiener theorem, In: Theory of Functions and Applications. Collection of Works dedicated to the Memory of Mkhitar M. Djrbashian, Louys Publishing House, Yerevan, pp. 193–196 (1995)Google Scholar
  24. 24.
    Tuan, V.K.: On the range of the Hankel and extended Hankel transforms. J. Math. Anal. Appl. 209, 460–478 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tuan, V.K.: Paley–Wiener-type theorems. Fract. Calc. Appl. Anal. 2, 135–143 (1999)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Tuan, V.K.: On the supports of functions. Numer. Funct. Anal. Optim. 20, 387–394 (1999)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tuan, V.K.: Paley–Wiener and Boas theorems for singular Sturm–Liouville integral transforms. Adv. Appl. Math. 29, 563–580 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tuan, V.K.: A real-variable Paley–Wiener theorem for the Dunkl transform, Abstract and Applied Analysis, 365–371. World Sci. Publ, River Edge, NJ (2004)Google Scholar
  29. 29.
    Tuan, V.K., Zayed, A.I.: Paley-Wiener-type theorem for a class of integral transforms arising from a singular Dirac system. Z. Anal. Anwendungen 19, 695–712 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tuan, V.K., Zayed, A.I.: Generalization of a theorem of Boas to a class of integral transforms. Results Math. 38, 362–376 (2000)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yang, Y., Qian, T.: Schwarz lemma in Euclidean spaces. Complex Var. Elliptic Eq. 51(7), 653–659 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

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