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Journal of Fourier Analysis and Applications

, Volume 20, Issue 4, pp 715–750 | Cite as

Discrete Hardy Spaces

  • Paula CerejeirasEmail author
  • Uwe Kähler
  • Min Ku
  • Frank Sommen
Article

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.

Keywords

Discrete Dirac operator Discrete Cauchy transform Discrete monogenic functions Hardy space 

Mathematics Subject Classification

Primary 44A15 Secondary 42A38 42C40 

Notes

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.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Paula Cerejeiras
    • 1
    Email author
  • Uwe Kähler
    • 1
  • Min Ku
    • 1
  • Frank Sommen
    • 2
  1. 1.Department of Mathematics, Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal
  2. 2.Clifford Research Group, Department of Mathematical AnalysisGhent UniversityGhentBelgium

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