Skip to main content
SpringerLink
Log in

Discrete Hardy Spaces

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy spaces. Hereby, we focus on the 3D-case with the generalization to the n-dimensional case being straightforward.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brackx, F., De Schepper, H., Sommen, F.: Discrete clifford analysis: a germ of function theory. In: Sabadini, I., et al. (eds.) Hypercomplex Analysis, pp. 37–53. Birkhäuser, Basel (2009)

    Google Scholar 

  2. Cerejeiras, P., Faustino, N., Vieira, N.: Numerical clifford analysis for nonlinear schrödinger problem. Numer. Methods Partial Differ. Equ. 24(4), 1181–1202 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Delanghe, R., Sommen, F., Soucek, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic, Dordrecht (1992)

    Book  MATH  Google Scholar 

  4. De Ridder, H., De Schepper, H., Kähler, U., Sommen, F.: Discrete function theory based on skew weyl relations. Proc. Am. Math. Soc. 138(9), 3241–3256 (2010)

    Article  MATH  Google Scholar 

  5. Faustino, N., Kähler, U.: Fischer decomposition for difference dirac operators. Adv. Appl. Clifford Algebras 17, 37–58 (2007)

    Article  MATH  Google Scholar 

  6. Faustino, N., Kähler, U., Sommen, F.: Discrete dirac operators in clifford analysis. Adv. Appl. Clifford Algebras 17(3), 451–467 (2007)

    Article  MATH  Google Scholar 

  7. Faustino, N., Gürlebeck, K., Hommel, A., Kähler, U.: Difference potentials for the navier–stokes equations in unbounded domains. J. Differ. Equ. Appl. 12(6), 577–595 (2006)

    Article  MATH  Google Scholar 

  8. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)

    MATH  Google Scholar 

  9. Forgy, E., Schreiber, U.: Discrete Differential Geometry on Causal Graphs. arXiv:math-ph/0407005v1 (2004) (preprint)

  10. Gilbert, J.E., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  11. Gürlebeck, K., Hommel, A.: On finite difference Dirac operators and their fundamental solutions. Adv. Appl. Clifford Algebras 11(S2), 89–106 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford calculus for physicists and engineers. Wiley, Chichester (1997)

    MATH  Google Scholar 

  13. Hommel, A.: Fundamentallösungen partieller Differentialoperatoren und die Lösung diskreter Randwertprobleme mit Hilfe von Differenzenpotentialen. PhD thesis, Bauhaus-Universitaät Weimar, Germany (1998)

  14. Kanamori, I., Kawamoto, N.: Dirac-Kaehler fermion from clifford product with noncommutative differential form on a lattice. Int. J. Mod. Phys. A19, 695–736 (2004)

    Article  MathSciNet  Google Scholar 

  15. Li, C., McIntosh, A., Qian, T.: Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces. Rev. Mat. Iberoamericana 10, 665–721 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lovasz, L.: Discrete Analytic Functions: An Exposition, Surveys in Differential Geometry, vol. IX, pp. 1–44. International Press, Somerville (2004)

    Google Scholar 

  17. McIntosh, A.: Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains. In: Ryan, J. (ed.) Clifford Algebras in Analysis and related Topics, pp. 33–88. CRC Press, Boca Raton (1996)

    Google Scholar 

  18. Mitrea, M.: Clifford Wavelets, Singular Integrals, and Hardy Spaces. Springer, Berlin (1994)

    MATH  Google Scholar 

  19. Ryabenskij, V.S.: The Method of Difference Potentials for Some Problems of Continuum Mechanics. Nauka, Moscow (1984). (Russian)

    Google Scholar 

  20. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  21. Thomée, V.: Discrete interior Schauder estimates for elliptic difference operators. SIAM J. Numer. Anal. 5, 626–645 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vaz, J.: Clifford-like calculus over lattices. Adv. Appl. Clifford Algebra 7(1), 37–70 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014. The third author is the recipient of a Postdoctoral Foundation from FCT (Portugal) under Grant No. SFRH/BPD/74581/2010.

Author information

Authors and Affiliations

  1. Department of Mathematics, Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal

    Paula Cerejeiras, Uwe Kähler & Min Ku

  2. Clifford Research Group, Department of Mathematical Analysis, Ghent University, Ghent, Belgium

    Frank Sommen

Authors
  1. Paula Cerejeiras
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uwe Kähler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Min Ku
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Frank Sommen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paula Cerejeiras.

Additional information

Communicated by Chris Heil.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cerejeiras, P., Kähler, U., Ku, M. et al. Discrete Hardy Spaces. J Fourier Anal Appl 20, 715–750 (2014). https://doi.org/10.1007/s00041-014-9331-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-014-9331-8

Keywords

Mathematics Subject Classification

Search

Navigation