The approximation formulas for integral operators which have been considered previously provide certain approximation orders up to a small saturation term, if the density is sufficiently smooth on the whole space. If the density has support contained in \(\Omega \subset {\mathbb R}^n\) and is different from zero at the boundary \(\partial \Omega \) , then the direct application of the method described in the preceding Chapters does not give good approximations, in general. In this Chapter we show how this difficulty can be overcome and how our method can be applied also if the densities are supported over half-spaces or hyper-rectangles of \({\mathbb R}^n\) .