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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Generalization of two results of the theory of uniform distribution

Author: Petko D. Proĭnov
Journal: Proc. Amer. Math. Soc. 95 (1985), 527-532
MSC: Primary 11K38
MathSciNet review: 810157
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Abstract: For a sequence $ {x_1}, \ldots ,{x_N}$ of points in $ [0,1]$ and a sequence $ {p_1}, \ldots ,{p_N}({p_1} + {p_2} + \cdots + {p_N} = 1)$ of nonnegative numbers, define the distribution function

$\displaystyle g(x) = x - \sum\limits_{{x_k} < x} {{p_k}.} $

Let $ \varphi $ be an increasing function on $ [0,1]$ and $ \varphi (0) = 0$. The main result of the paper is

$\displaystyle F({D_N}) \leqslant \int_0^1 {\varphi \left( {\left\vert {g(x)} \right\vert} \right)} dx \leqslant \varphi ({D_N}),$

where $ {D_N}$ is the supremum norm of $ g$ on $ [0,1]$ and $ F$ is the antiderivative of $ \varphi $ with $ F(0) = 0$. This result generalizes and improves an estimate of Niederreiter [1] for the $ {L^2}$ discrepancy of the sequence $ {x_1}, \ldots ,{x_N}$. Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.

References [Enhancements On Off] (What's this?)

  • [1] H. Niederreiter, Application of diophantine approximation to numerical integration, Diophantine Approximation and its Applications (C. F. Osgood, ed.), Academic Press, New York, 1973, pp. 129-199. MR 0357357 (50:9825)
  • [2] I. M. Sobol, Multidimensional quadrature formulae and Haar functions, Nauka, Moscow, 1969. MR 0422968 (54:10952)
  • [3] P. D. Proinov, Note on the convergence of the general quadrature process with positive weights, Constructive Function Theory'77 (Bl. Sendov and D. Vačov, eds.), Sofia, 1980, pp. 121-125.

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