Projnov, Petko D. Symmetrization of the van der Corput generalized sequences. (English) Zbl 0654.10049 Proc. Japan Acad., Ser. A 64, No. 5, 159-162 (1988). The author considers distribution properties of the van der Corput generalized sequence \(\sigma =(\phi (n))_ 0^{\infty}\) to the base \((r_ j)_ 1^{\infty}\) as defined by Faure. For example he shows that: When \(\sum^{N}_{j=1}r_ j^ 2=O(N)\) then the order of magnitude of the diaphony of this sequence is as small as possible namely \(N^{-1}(\log N)^{1/2}.\) Analogously the order of magnitude of the \(L^ 2\)-discrepancy of any symmetrical sequence produced by \(\sigma\) is as small as possible namely again \(N^{-1} (\log N)^{1/2}.\) Reviewer: G.Larcher Cited in 1 ReviewCited in 12 Documents MSC: 11K38 Irregularities of distribution, discrepancy 11K06 General theory of distribution modulo \(1\) Keywords:uniform distribution; van der Corput generalized sequence; order of magnitude; diaphony; \(L^ 2\)-discrepancy; symmetrical sequence × Cite Format Result Cite Review PDF Full Text: DOI References: [1] H. Davenport: Note on irregularities of distribution. Mathematika, 3, 131-135 (1956). · Zbl 0073.03402 · doi:10.1112/S0025579300001807 [2] H. Faure: Discrepance de suites associees a un systeme de numeration (en dimension un). Bull. Soc. Math. France, 109, 143-182 (1981). · Zbl 0488.10052 [3] H. Faure: Discrepance quadratique de suites infinies en dimension un. Proc. Conf. on Number Theory, University of Laval, Quebec, Canada, pp. 22-25 (1987). [4] L. Kuipers and H. Niederreiter: Uniform Distribution of Sequences. Wiley (1974). · Zbl 0281.10001 [5] P. D. Proinov: On the L2 discrepancy of some infinite sequences. Serdica Bulg. Math. Publ., 11, 3-12 (1985). · Zbl 0584.10033 [6] P. D. Proinov: On irregularities of distribution. C. R. Acad. Sci. Bulgare, 39, no. 9, 31-34 (1986). [7] P. D. Proinov and V. S. Grozdanov: Symmetrization of the van der Corput-Halton sequence, ibid., 40, no. 8, 5-8 (1987). · Zbl 0621.10035 [8] K. F. Roth: On irregularities of distribution. Mathematika, 1, 73-79 (1954). · Zbl 0057.28604 · doi:10.1112/S0025579300000541 [9] I. M. SoboF: Multidimensional Quadrature Formulae and Haar Functions. Moscow (1969) (in Russian). [10] P. Zinterhof and H. Stegbuchner: Trigonometrische Approximation mit Gleichverteilungsmethoden. Studia Sci. Math. Hung., 13, 273-289 (1978). · Zbl 0421.42001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.