Skip to main content

Subharmonic Functions on Discrete Structures

  • Conference paper
Harmonic Analysis, Signal Processing, and Complexity

Part of the book series: Progress in Mathematics ((PM,volume 238))

Abstract

We define the Laplacian on an arbitrary set with a not necessarily symmetric weight function and discuss the Dirichlet problem and other classical topics in this setting.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • M. Baker and R. Rumely (2004), Harmonic analysis on metrized graphs, manuscript; available online from arxiv.org/abs/math/0407427.

    Google Scholar 

  • C. Blanc (1939–1940), Une interprĂ©tation Ă©lĂ©mentaire des thĂ©orèmes fondamentaux de M. Nevanlinna, Comm. Math. Helv., 12, 153–163.

    Article  MathSciNet  Google Scholar 

  • C. BrechbĂĽhler, G. Gerig, and O. KĂĽbier (1995), Parametrization of closed surfaces for 3-D shape description, Comput. Vision Image Understanding, 6-2, 154–170.

    Article  Google Scholar 

  • R. H. Burkhardt (1997), Asympototic expansion of the free-space Green’s function for the discrete 3-D Poisson equation, SIAM J. Sci. Comput., 18, 1142–1162.

    Article  MathSciNet  Google Scholar 

  • R. J. Duffin (1953), Discrete potential theory, Duke Math. J., 20, 233–251.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Chung and S.-T. Yau (2000), Discrete Green’s functions, J. Combin. Theory Ser. A, 91, 191–214.

    Article  MATH  MathSciNet  Google Scholar 

  • C. Favre and M. Jonsson (2004), The Valuative Tree, Lecture Notes in Mathematics 1853, Springer-Verlag, New York.

    MATH  Google Scholar 

  • J. Ferrand (1944), Fonctions prĂ©harmoniques et fonctions prĂ©holomorphes, Bull. Sci. Math., 68, 152–180.

    MATH  MathSciNet  Google Scholar 

  • H.-Y. Guo and K. Wu (2003), On variations in discrete mechanics and field theory, J. Math. Phys., 44, 5978–6004.

    Article  MATH  MathSciNet  Google Scholar 

  • R. Kenyon (2002), The Laplacian and Dirac operators on critical planar graphs, Invent. Math., 150, 409–439.

    Article  MATH  MathSciNet  Google Scholar 

  • S. P. Novikov and I. A. Dynnikov (1997), Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds, Uspekhi Mat. Nauk, 52, 1058–1116 (in Russian); Russian Math. Surveys, 52 (1997), 1057–1116 (in English).

    Google Scholar 

  • H. B. Phillips and N. Wiener (1923), Nets and the Dirichlet problem, J. Math. Phys., 2,105–124; also available in Norbert Wiener: Collected Works, Vol. I, MIT Press, Cambridge, MA, London, 1976, 333–354.

    Google Scholar 

  • O. Weistrand (2004), Shape approximation of digital surfaces homeomorphic to a sphere, manuscript, Uppsala University, Uppsala, Sweden.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Additional information

Dedicated to Carlos Berenstein and Daniele Struppa.

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Boston

About this paper

Cite this paper

Kiselman, C.O. (2005). Subharmonic Functions on Discrete Structures. In: Sabadini, I., Struppa, D.C., Walnut, D.F. (eds) Harmonic Analysis, Signal Processing, and Complexity. Progress in Mathematics, vol 238. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4416-4_6

Download citation

Publish with us

Policies and ethics