Uniform Distribution and Numerical Integration

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

$$ \begin{gathered} {\Delta_{10}}f\left( {{x_i},y} \right) = f({x_{i + 1}},y) - f({x_i},y), \hfill \\ {\Delta_{01}}f(x,{y_i}) = f(x,{y_{i + 1}}) - f(x,{y_1}), \hfill \\ {\Delta_{11}}f({x_i},{y_i}) = f({x_i},y{}_i) - f({x_{i + 1}},{y_i}) - f({x_i},{y_{y + 1}}) + f({x_{i + 1}},{y_{i + 1}}) \hfill \\ \end{gathered} $$

.