manuscripta mathematica

, Volume 95, Issue 1, pp 137–147 | Cite as

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

  • Holger Boche


The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous functiong 1 εC 0[0,1]2 with support in the rectangle [0,1]×[0,1/2] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1]×[1/2,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer.

Mathematics Subject Classification (1991)

41A05 94A05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Boche, H.:Über das Lokalisierungsprinzip bei mehrdimensionalen Shannonschen Abtastreihen. IEE-10-1995, TU DresdenGoogle Scholar
  2. [2]
    Boche, H.:Neue Untersuchungen zu den Shannonschen Abtastreihen. Preprint, FSU-Jena, 1995Google Scholar
  3. [3]
    Butzer, P.L.:Persönliche Mitteilung, RWTH-Aachen, 1995Google Scholar
  4. [4]
    Butzer, P.L., Splettstößer, W., Stens, R.:The Sampling Theorem and Linear Prediction in Signal Analysis. Jber. Deutsch. Math.-Vereinigung90, 1–70 (1988)zbMATHGoogle Scholar
  5. [5]
    Butzer, P.L., Stens, R.:Sampling Theory for not necessarily band-limited Functions. SIAM Review, March 1992, Vol.34, No. 1Google Scholar
  6. [6]
    Butzer, P.L.:A survey of Whittaker-Shannon sampling theorem and some of its extensions. J. Math. Res. Exposition,3, p. 185–212 (1983)MathSciNetGoogle Scholar
  7. [7]
    Dryanow, D.P.:On the convergence and saturation problem of a sequence of discrete linear Operators of exponential type in L p (−∞, ∞) Spaces. Acta Math. Hung.49 (1–2), 103–127 (1987)CrossRefGoogle Scholar
  8. [8]
    de la Vallée Poussin, J.-Ch.:Sur la convergence des formules d’interpolation entre ordonnées équidistantes. Bull. Acad. roy. Belg.1908, 319–410Google Scholar
  9. [9]
    Jerri, A.:The Shannon sampling theorem — its varios extensions and applications: a tutorial review. Proc. IEEE65, 1565–1596 (1977)zbMATHCrossRefGoogle Scholar
  10. [10]
    Marks, R.J.:Introduction to Shannon Sampling and Interpolation Theory. Springer Texts in Electrical Engineering, New York: Springer-Verlag, 1991Google Scholar
  11. [11]
    Marks, R.J. (ed.):Advanced Topics in Shannon Sampling and Interpolation Theory. Springer Texts in Electrical Engineering, New York: Springer-Verlag, 1993Google Scholar
  12. [12]
    Ries, S., Stens, R.L.:A localization principle for the approximation by sampling series. In: Theory of Approximation of Functions (Russian), Proc. Intern. Conf. Kiev 1983, Eds.: N.P. Korneicuk-S.B. Steckin-S.A. Teljakovskii, Izdat Nauka, Moskau, 1987, pp. 507–510Google Scholar
  13. [13]
    Stens, R.L.:Persönliche Mitteilung. RWTH-Aachen, 1995Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Holger Boche
    • 1
    • 2
    • 3
  1. 1.Mathematisches Institut Fakultät für Mathematik und InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbHBerlinGermany
  3. 3.ETH Zürich, Institut für Kommunikationstechnik, ETH-ZentrumZürich

Personalised recommendations