# 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

Uniform Distribution Error Term Number Theory Monotonic Function Variation Versus
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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