Uniform Distribution and Numerical Integration

  • Hua Loo Keng
  • Wang Yuan

Abstract

Let
$$ \begin{gathered} 0 = {x_0} < {x_i} < ... < {x_l} = 1, \hfill \\ 0 = {y_0} < {y_1} < ... < {y_m} = 1 \hfill \\ \end{gathered} $$
(σ)
be any division of G2. Let f(x,y) be a function defined on G2 and
$$ \begin{gathered} {\Delta_{10}}f\left( {{x_i},y} \right) = f({x_{i + 1}},y) - f({x_i},y), \hfill \\ {\Delta_{01}}f(x,{y_i}) = f(x,{y_{i + 1}}) - f(x,{y_1}), \hfill \\ {\Delta_{11}}f({x_i},{y_i}) = f({x_i},y{}_i) - f({x_{i + 1}},{y_i}) - f({x_i},{y_{y + 1}}) + f({x_{i + 1}},{y_{i + 1}}) \hfill \\ \end{gathered} $$
.

Keywords

Estima 

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Notes

  1. The definition for a function of bounded variation was given by M. Krause [1] and 6. H. Hardy [1] (Cf. also C. E. Adams, and J. A. Clarkson [1,2] and S. K. Zaremba [2]).Google Scholar
  2. Theorem 5.3 was proved by J. F. Koksma [1] for s = 1 and generalized to s > 1 by E. EOawka [1] (Cf. also E. M. Sobol [1] for the class of functions Ls).Google Scholar
  3. Theorem 5.6: Cf. N. S. Bahvalov [1].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Science Press. Beijing 1981

Authors and Affiliations

  • Hua Loo Keng
    • 1
  • Wang Yuan
    • 1
  1. 1.Institute of MathematicsAcademia SinicaBeijingThe People’s Republic of China

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