In this paper we introduce multiresolution analysis (MRA) algorithms intended to be used in scientific visualization, and based on a non-nested set of approximating spaces. The need for non nested spaces arises from the fact that the required scaling functions do not fulfill any refinement equation. Therefore we introduce in the first part the concept of approximated refinement equation, that allows to generalize the filter bank and exact reconstruction algorithms. The second part shows how this concept enables to define a MRA scheme for piecewise constant data defined on an arbitrary planar or spherical triangular mesh. The ability to deal with arbitrary triangular meshes, without subdivision connectivity, can be achieved only through the use of non nested approximating spaces, as introduced in the first part.