  This paper considers several approximate operators used in a particle method
based on a Voronoi diagram. We introduce and study our approximate operators on
gradient and Laplace operators. We derive error estimates for these approximate
operators by applying our weight functions. The key idea of deriving our error
estimates is to divide the integration region into a ring-shaped area and some
areas. In the Appendix, we give an example application of the main results of
this paper.
