[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

On the Supports of Functions Associated to the Radially Deformed Fourier Transform

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra \({\mathfrak {osp}}(1|2)\) in terms of a so-called radially deformed Dirac operator \({\mathbf {D}}\) depending on a deformation parameter c such that for \(c=0\) the classical Dirac operator is reobtained. In this paper, several versions of the Paley–Wiener theorems for this radially deformed Fourier transform are investigated, which characterize the supports of functions associated to this generalized Fourier transform in Clifford analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andersen, N.B.: Real Paley–Wiener theorems. Bull. Lond. Math. Soc. 36, 504–508 (2004)

    Article  MathSciNet  Google Scholar 

  2. Andersen, N.B., de Jeu, M.: Real Paley–Wiener theorems and local spectral radius formulas. Trans. Am. Math. Soc. 362, 3613–3640 (2010)

    Article  MathSciNet  Google Scholar 

  3. Bang, H.H.: A property of infinitely differentiable functions. Proc. Am. Math. Soc. 108, 73–76 (1990)

    Article  MathSciNet  Google Scholar 

  4. Bang, H.H.: Functions with bounded spectrum. Trans. Am. Math. Soc. 347, 1067–1080 (1995)

    Article  MathSciNet  Google Scholar 

  5. Ben Saïd, S., Kobayashi, T., Ørsted, B.: Laguerre semigroup and Dunkl operators. Compos. Math. 148(4), 1265–1336 (2012)

    Article  MathSciNet  Google Scholar 

  6. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)

    MATH  Google Scholar 

  7. De Bie, H., Ørsted, B., Somberg, P., Souček, V.: The Clifford deformation of the Hermite semigroup. SIGMA Symmetry Integrability Geom. Methods Appl., vol. 9, Paper 010 (2013)

  8. De Bie, H.: Clifford algebras, Fourier transforms, and quantum mechanics. Math. Methods Appl. Sci. 35(18), 2198–2228 (2012)

    Article  MathSciNet  Google Scholar 

  9. De Bie, H., Ørsted, B., Somberg, P., Souček, V.: Dunkl operators and a family of realizations of \(\mathfrak{osp}(1|2)\). Trans. Am. Math. Soc. 364(7), 3875–3902 (2012)

    Article  MathSciNet  Google Scholar 

  10. De Bie, H., De Schepper, N., Eelbode, D.: New results on the radially deformed Dirac operator. Complex Anal. Oper. Theory 11(6), 1283–1307 (2017)

    Article  MathSciNet  Google Scholar 

  11. Delanghe, R., Sommen, F., Soucek, V.: Clifford Algebra and Spinor Valued Functions, A Function Theory for Dirac Operator. Kluwer, Dordrecht (1992)

    Book  Google Scholar 

  12. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)

    Article  MathSciNet  Google Scholar 

  13. Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  14. Heckman, G.J.: A remark on the Dunkl differential-difference operators. In: Barker, W. (ed.) Harmonic Analysis on Reductive Groups, Progress in Mathematics, vol. 101, pp. 181–191. Birkhäuser, Basel (1991)

    Chapter  Google Scholar 

  15. Kobayashi, T., Mano, G.: The inversion formula and holomorphic extension of the minimal representation of the conformal group. In: Li, J.S., Tan, E.C., Wallach, N., Zhu, C.B. (eds.) Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory: in Honor of Roger Howe, pp. 159–223. World Scientific, Singapore (2007). (cf. math.RT/0607007)

    Google Scholar 

  16. Li, S., Leng, J., Fei, M.: Paley–Wiener-type theorems for the Clifford-Fourier transform. Math. Methods Appl. Sci. 42(18), 6101–6113 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  17. Li, S., Leng, J., Fei, M.: Spectrums of functions associated to the fractional Clifford-Fourier transform. Adv. Appl. Clifford Algebra 30(1), 6 (2020)

    Article  MathSciNet  Google Scholar 

  18. 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, pp. 193–196. Louys Publishing House, Yerevan (1995)

  19. Tuan, V.K.: Paley–Wiener-type theorems. Fract. Calc. Appl. Anal. 2, 135–143 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Yang, Y., Qian, T.: Schwarz lemma in Euclidean spaces. Complex Var. Elliptic Equ. 51(7), 653–659 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Southwest Minzu University, Chengdu, 610041, People’s Republic of China

    Shanshan Li

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, People’s Republic of China

    Jinsong Leng & Minggang Fei

Authors
  1. Shanshan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinsong Leng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Minggang Fei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minggang Fei.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors were partially supported by “the Fundamental Research Funds for the Central Universities” with no. ZYGX2019J091.

This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Leng, J. & Fei, M. On the Supports of Functions Associated to the Radially Deformed Fourier Transform. Adv. Appl. Clifford Algebras 30, 47 (2020). https://doi.org/10.1007/s00006-020-01067-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-020-01067-7

Keywords

Mathematics Subject Classification

Search

Navigation