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The exact order of generalized diaphony and multidimensional numerical integration

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  09 April 2009

Vsevolod F. Lev
Vsevolod F. Lev
Affiliation:
Department of Mathematics, The University of Georgia, Athens GA 30605, USA e-mail: [email protected]
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Abstract

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For a point set in the multidimensional unit torus we introduce an Lk-measure of uniformity of distribution, which for k=2 reduces to diaphony (and thus in this case essentially coincides with Weyl L2-discrepancy). For k ∈ [1, 2] we establish a sharp asymptotic for this new measure as the number of points of the set tends to infinity. Upper and lower-bound estimates are given also for k >2.

Keywords

Diaphonydiscrepancyuniform distribution

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy 11K70: Harmonic analysis and almost periodicity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 1 , February 1999 , pp. 1 - 17
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bykowski, V. A., ‘On the right order of error of optimal cubature formulas in the spaces with dominating derivation and L 2-discrepancy of nets.’ (Dalnevost. Science Center of the USSR Acad. of Sciences, Vladivostok), preprint (in Russian).Google Scholar
[2]Chaix, H. and Faure, H., ‘Discrépance et diaphonie en dimension un’, Acta Arith. 63 (1993), 103–141.CrossRefGoogle Scholar
[3]Chaix, H., ‘Discrépance et diaphonie des suites de van der Corput généralisées. III’, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 755–758.Google Scholar
[4]Dobrovolsky, N. M., Number-theoretic nets and their applications (Diss. Kand. Phis.-math. nauk, Tula, 1984) (in Russian).Google Scholar
[5]Frolov, K. K., ‘Upper-bound error estimates on classes of functions’, Dokl. Akad. Nauk SSSR 231 (1976), 818–821. (in Russian)Google Scholar
[6]Korobov, N. M., Number-theoretic methods in numerical analysis (Fizmatgiz, Moscow, 1963) (in Russian).Google Scholar
[7]Lev, V. F., ‘Translations of nets and relationship between supreme and Lk-discrepancies’, Acta Math. Hungar. 70 (1996), 1–12.CrossRefGoogle Scholar
[8]Lev, V. F., ‘On quadrature formulas for classes with restrictions on Fourier coefficients’, Deposited to VINITI (USSR) 27.8.1987, no. 6294-B87 (in Russian).Google Scholar
[9]Lev, V. F., ‘Hyperbolic zeta-function of general form’, Deposited to VINITI (USSR) 27.8.1987, no. 6296-B87 (in Russian).Google Scholar
[10]Lev, V. F., ‘Diaphony and L2-discrepancies of multidimensional nets’, Mat. Zametki 47 (1990), 45–54 (in Russian).Google Scholar
English translation in Math. Notes 47 (1990), 556–564.CrossRefGoogle Scholar
[11]Lev, V. F., ‘Hyperbolic zeta-function of general form’, Mat. Zametki 48 (1990), 148–150 (In Russian).Google Scholar
[12]Lev, V. F., ‘Quadrature formulas for classes with restrictions on Fourier coefficients’, Mat. Zametki 49 (1991), 219–221 (in Russian).Google Scholar
English translation in Math. Notes 49 (1991), 219–221.CrossRefGoogle Scholar
[13]Lev, V. F., ‘On two versions of L2-discrepancy and geometrical interpretation of diaphony’, Acta Math. Hungar. 69 (1995), 281–300.CrossRefGoogle Scholar
[14]Paghs, G., ‘Van der Corput sequences, Kakutani transforms and one-dimensional numerical integration’, J. Comput. Appl. Math. 44 (1992), 21–39.CrossRefGoogle Scholar
[15]Proinov, P. D., and Grozdanov, V. S, ‘On the diaphony of the van der Corput-Halton sequence’, J. Number Theory 30 (1988), 94–104.CrossRefGoogle Scholar
[16]Proinov, P. D. and Atanassov, E. Y., ‘On the distribution of the van der Corput generalized sequences’, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 895–900.Google Scholar
[17]Roth, K. F., ‘On irregularities of distribution. I’, Mathematika 1 (1954), 73–79.CrossRefGoogle Scholar
[18]Weyl, H., ‘Über die Gleichverteilung von Zahlen mod. Eins’, Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar
[19]Xiao, Y.-J., ‘Suites équiréparties associées aux automorphismes du tore’, C. R. Acad. Sci. Paris Sér I Math. 311 (1990), 579–582.Google Scholar
[20]Zinterhof, P., ‘Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden’, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 185 (1976), 121–132.Google Scholar