  In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte
Carlo (QMC) integration. QMC integration of a function
$f\colon[0,1)^s\rightarrow \mathbb{R}$ by a finite point set
$\mathcal{P}\subset [0,1)^s$ is the approximation of the integral
$I(f):=\int_{[0,1)^s}f(\mathbf{x})\,d\mathbf{x}$ by the average
$I_{\mathcal{P}}(f):=\frac{1}{|\mathcal{P}|}\sum_{\mathbf{x} \in
\mathcal{P}}f(\mathbf{x})$. We treat a certain class of point sets
$\mathcal{P}$ called digital nets. A Koksma-Hlawka type inequality is an
inequality bounding the integration error
$\text{Err}(f;\mathcal{P}):=I(f)-I_{\mathcal{P}}(f)$ by a bound of the form
$|\text{Err}(f;\mathcal{P})|\le C\cdot \|f\|\cdot D(\mathcal{P})$. We can
obtain a Koksma-Hlawka type inequality by estimating bounds on
$|\hat{f}(\mathbf{k})|$, where $\hat{f}(\mathbf{k})$ is a generalized Fourier
coefficient with respect to the Walsh system. In this paper we prove bounds on
Walsh coefficients $\hat{f}(\mathbf{k})$ by introducing an operator called
`dyadic difference' $\partial_{i,n}$. By converting dyadic differences
$\partial_{i,n}$ to derivatives $\frac{\partial }{\partial x_i}$, we get a new
bound on $|\hat{f}(\mathbf{k})|$ for a function $f$ whose mixed partial
derivatives up to order $\alpha$ in each variable are continuous. This new
bound is smaller than the known bound on $|\hat{f}(\mathbf{k})|$ under some
condition. The new Koksma-Hlawka inequality is derived using this new bound on
the Walsh coefficients.
