Editing Low-discrepancy sequence (section)

==The ''L<sup>2</sup>'' version of the Koksma–Hlawka inequality==

Applying the [[Cauchy–Schwarz inequality]] for integrals and sums to the Hlawka–Zaremba identity, we obtain an <math>L^2</math> version of the Koksma–Hlawka inequality:

: <math>
\left|\frac{1}{N} \sum_{i=1}^N f(x_i)
      - \int_{\bar I^s} f(u)\,du\right|\le
\|f\|_d \operatorname{disc}_d (\{t_i\}),
</math>

where

:<math>
\operatorname{disc}_d(\{t_i\})=\left(\sum_{\emptyset\neq u\subseteq D}
\int_{[0,1]^{|u|}} \operatorname{disc}(x_u,1)^2 \, dx_u\right)^{1/2}
</math>

and

:<math>
\|f\|_d = \left(\sum_{u\subseteq D}
\int_{[0,1]^{|u|}}
\left|\frac{\partial^{|u|}}{\partial x_u} f(x_u,1)\right|^2 dx_u\right)^{1/2}.
</math>

<math>L^2</math> discrepancy has a high practical importance because fast explicit calculations are possible for a given point set. This way it is easy to create point set optimizers using <math>L^2</math> discrepancy as criteria.