1. bookVolumen 16 (2021): Heft 1 (June 2021)
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Zeitschrift
eISSN
2309-5377
Erstveröffentlichung
30 Dec 2013
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The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ s

Online veröffentlicht: 30 Oct 2021
Volumen & Heft: Volumen 16 (2021) - Heft 1 (June 2021)
Seitenbereich: 71 - 92
Eingereicht: 02 Feb 2021
Akzeptiert: 10 May 2021
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2309-5377
Erstveröffentlichung
30 Dec 2013
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
Abstract

In the present paper the author uses the function system Γsconstructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.

MSC 2010

[1] CHRESTENSON, H. E.: A class of generalized Walsh functions, Pacific J. Math. 5 (1955), 17–31.10.2140/pjm.1955.5.17 Search in Google Scholar

[2] DRMOTA, M.—TICHY, R. F.: Sequences, Discrepancies and Applications. In: Lecture Notes in Math. Vol. 1651, Springer-Verlag, Berlin, 1997. Search in Google Scholar

[3] ERDŐS, P.—TURÁN, P.: On a problem in the theory of uniform distribution, I, II. Indag. Math. 10 (1948), 370–378; 406–413. Search in Google Scholar

[4] GROZDANOV, V.—PETROVA, TS.: The function system Γs and its applications to the theory of Quasi-Monte Carlo integration and uniformly distributed sequen (to appear). Search in Google Scholar

[5] HELLEKALEK, P.: General discrepancy estimates: the Walsh function system, Acta Arith. 67 (1994), 209–218.10.4064/aa-67-3-209-218 Search in Google Scholar

[6] HELLEKALEK, P.: A general discrepancy estimate based on p-adic arithmetics, Acta Arith. 139 (2009), no. 2, 117–129. Search in Google Scholar

[7] HELLEKALEK, P.—NIEDERREITER, H.: Constructions of uniformly distributed sequences using the b−adic method, Unif. Distrib. Theory 6 (2011), no. 1, 185–200. Search in Google Scholar

[8] KOKSMA, J. F.: Some Theorems on Diophantine Inequalities, Scriptum, Vol. 5, Math. Centrum Amsterdam, 1950. Search in Google Scholar

[9] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences, John Wiley and Sons, New York, 1974. Search in Google Scholar

[10] LARCHER, G.—NIEDERREITER, H.—SCHMID, W. CH.: Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration, Monatsh. Math. 121 1996, 231–253.10.1007/BF01298952 Search in Google Scholar

[11] NIEDERREITER, H.: Random Number Generator and Quasi-Monte Carlo Methods. SIAM, Philadelphia, 1992.10.1137/1.9781611970081 Search in Google Scholar

[12] VILENKIN, N. YA.: On a class of complete orthonormal systems, Izv. Akad. Nauk. SSSR Ser. Math. 11 (1947), 363–400. (In Russian) Search in Google Scholar

[13] WALSH, J. L.: A closed set of normal orthogonal functions, Amer. J. Math. 45 1923, 5–24.10.2307/2387224 Search in Google Scholar

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