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manuscripta mathematica

, Volume 1, Issue 3, pp 293–306 | Cite as

Diskrepanz in kompakten abelschen Gruppen II

  • Harald Niederreiter
Article

Abstract

This is a continuation of the paper [6]. The concept of discrepancy of a sequence in a compact abelian group with second countability axiom is now investigated in its relation to problems of distribution and approximation. Some fundamental inequalities of the theory of uniform distribution mod 1 can be transferred to the general case, e.g. Koksma's inequality. The crucial point in the generalization of Koksma's inequality is the suitable definition of total variation. Another version of this inequality with an upper bound in terms of the Fourier coefficients is presented. Aardenne-Ehrenfest's theorem is valid for non totally disconnected groups.

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Literatur

  1. [1]
    AARDENNE-EHRENFEST, T. van: On the impossibility of a just distribution. Indagationes math. 11, 264–269 (1949).Google Scholar
  2. [2]
    BEER, S.: Zur Theorie der Gleichverteilung im p-adischen. Österreich. Akad. Wiss., math.-naturw.Kl., S.-Ber., Abt. II 176, 499–519 (1967).Google Scholar
  3. [3]
    CIGLER, J.-HELMBERG, G.: Neuere Entwicklungen der Theorie der Gleichverteilung. J. -ber. Deutsch. Math.- Verein. 64, 1–50 (1961).Google Scholar
  4. [4]
    HARDY, G.H.: On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Quart. J. Math., Oxford I. Ser. 37, 53–79 (1906).Google Scholar
  5. [5]
    HLAWKA, E.: Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. pura appl. IV. Ser. 54, 325–333 (1961).Google Scholar
  6. [6]
    HLAWKA, E.-NIEDERREITER, H.: Diskrepanz in kompakten abelschen Gruppen. Manuscripta math.Google Scholar
  7. [7]
    KOKSMA, J. F.: Ein allgemeiner Satz der Theorie der Gleichverteilung mod 1, Einige Integrale in der Theorie der Gleichverteilung mod 1 (holl.). Mathematica, Zutphen B 11, 7–11, 49–52 (1942).Google Scholar
  8. [8]
    KRAUSE, J. M.: Fouriersche Reihen mit zwei veränderlichen Größen. Ber. Verh. Sächs. Akad. Wiss., Leipzig, math.naturw.Kl. 55, 164–197 (1903).Google Scholar
  9. [9]
    WEIL, A.: L'intégration dans les groupes topologiques et ses applications, 2ème éd., Actualités Sci. et Ind. 869, 1145. Paris: Hermann&Cie. 1951.Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • Harald Niederreiter
    • 1
  1. 1.Mathematisches Institut der Universität WienWien IX

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