On Irregularities of Distribution and Approximate Evaluation of Certain Functions II

  • W. W. L. Chen
Part of the Progress in Mathematics book series (PM, volume 70)

Abstract

Let U =[0,1]. Suppose that g is a Lebesgue-integrable function, not necessarily bounded, in U 2, and that h is any function in U 2. Let P = P(N) be a distribution of N points in U 2 such that h(y) is finite for every y ∈ P. For x = (x1,x2) in U 2, let B(x) denote the rectangle consisting of all y = (y1,y2) in U 2 satisfying 0 < yl < x1 and 0 < y2 < x2, and write
$${\rm{Z}}\left[ {P;{\rm{h}}:B({\rm{x}})} \right] = \sum\limits_{y{\kern 1pt} \in {\kern 1pt} P{\kern 1pt} \cap B({\rm{x}})} {{\rm{h(y}}).}$$
(1)
Let μ denote the Lebesgue measure in U2, and write
$${\rm{D}}\left[ {P;{\rm{h;g}};B({\rm{x}})} \right] = {\rm{Z}}\left[ {P;{\rm{h}};B({\rm{x}})} \right] - {\rm{N}}\int\limits_{B({\rm{x}})} {{\rm{g}}({\rm{y}}){\rm{d}}\mu } .$$
(2)

Keywords

Ethod 

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References

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    [1].W. W. L. Chen, On irregularities of distribution and approximate evaluation of certain functions, to appear in Quarterly Journal of Mathematics (Oxford) 1985.Google Scholar
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    G. Halász, On Roth’s method in the theory of irregularities of point distributions, Recent progress in analytic number theory, vol. 2, pp. 79–94 (Academic Press, London, 1981).Google Scholar
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    W. M. Schmidt, Irregularities of distribution VII, Acta Arith., 21 (1972), 45–50.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • W. W. L. Chen
    • 1
  1. 1.Imperial CollegeLondonUK

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