Summary
The (weighted) dyadic diaphony is a measure for the irregularity of distribution modulo one of a sequence. Recently it has been shown that the (weighted) dyadic diaphony can be interpreted as the worst-case error for QMC integration in a certain Hilbert space of functions. In this paper we give upper bounds on the weighted dyadic diaphony of digital (t, s)-sequences over ℤ2.
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References
H.E. Chrestenson. A class of generalized Walsh functions. Pacific J. Math., 5:17–31, 1955.
J. Dick and F. Pillichshammer. Diaphony, discrepancy, spectral test and worst-case error. Math. Comput. Simulation, 70:159–171, 2005.
J. Dick and F. Pillichshammer. Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complexity, 21(2):149–195, 2005.
J. Dick and F. Pillichshammer. On the mean square weighted L2 discrepancy of randomized digital (t, m, s)-nets over ℤ2. Acta Arith., 117:371–403, 2005.
M. Drmota and R. F. Tichy. Sequences, discrepancies and applications. Lecture Notes in Mathematics. 1651. Berlin: Springer., 1997.
V. Grozdanov. The weighted b-adic diaphony. J. Complexity, 22:490–513, 2006.
P. Hellekalek and H. Leeb. Dyadic diaphony. Acta Arith., 80:187–196, 1997.
L. Kuipers and H. Niederreiter. Uniform distribution of sequences. Pure and Applied Mathematics. New York etc.: John Wiley & Sons., 1974.
P. Kritzer. Improved upper bounds on the star discrepancy of (t, m, s)-nets and (t, s)-sequences. J. Complexity, 22:336–347, 2006.
G. Larcher. Digital point sets: Analysis and application. Hellekalek, Peter (ed.) et al., Random and quasi-random point sets. New York, NY: Springer. Lect. Notes Stat., Springer-Verlag. 138, 167–222, 1998.
G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatsh. Math., 121:231–253, 1996.
K. Niederdrenk. Die endliche Fourierund Walsh-Transformation mit einer Einführung in die Bildverarbeitung. Braunschweig - Wiesbaden: Vieweg, 1982.
H. Niederreiter. Point sets and sequences with small discrepancy. Monatsh. Math., 104:273–337, 1987.
H. Niederreiter. Random number generation and quasi-Monte Carlo methods. CBMS-NSF Regional Conference Series in Applied Mathematics. 63. Philadelphia, SIAM, Society for Industrial and Applied Mathematics., 1992.
F. Pillichshammer. Dyadic diaphony of digital sequences. J. Théor. Nombres Bordx. (to appear), 2007.
G. Pirsic. Schnell konvergierende Walshreihen über Gruppen. Diplomarbeit, University of Salzburg, 1995.
O. Strauch and Š. Porubský. Distribution of sequences. A sampler. Schriftenreihe der Slowakischen Akademie der Wissenschaften 1. Bern: Peter Lang, 2005.
J. L. Walsh. A closed set of normal orthogonal functions. American J. Math., 45:5–24, 1923.
P. Zinterhof. Ü ber einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Sitzungsber. Osterr. Akad. Wiss. Math.-Natur. Kl. II, 185:121–132, 1976.
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Kritzer, P., Pillichshammer, F. (2008). The Weighted Dyadic Diaphony of Digital Sequences. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_32
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DOI: https://doi.org/10.1007/978-3-540-74496-2_32
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