Considered in this paper are the vector spaces of multiwavelets, functions defined on a small support, which are refined by halving intervals. A good history of past papers is presented to get the reader up to speed. By choosing basis elements for the refinement of a grid that are orthogonal to themselves because of their finite support, and that are orthogonal to the basis of the parent grid, a hierarchy of basis elements that are mutually orthogonal to each other can be built.

The formulas to generate these basis elements, starting from the Legendre polynomials, are developed. The objective is to form the matrix representation of some typical operators (transition matrices) acting on these spaces of functions, so that partial differential equations (PDEs), like the heat equation and Burgers equation, can be solved numerically, that is, integrated forward in time, using these matrix representations of the exponential of the space differential operator. These matrices are likely to be sparse, as the basis is orthogonal. The authors seek to adaptively refine the grid in areas needing higher accuracy of approximation.

The paper is heavy reading at times, since many of the variables need three sub- or superscripts to represent the numbering of basis elements within a refinement level. Some effort is expended to include the boundary conditions in the representations, by careful choice of basis elements. The paper finishes with the examples of the heat equation and Burgers equation in one space dimension. The results are impressive in terms of accuracy, and it is nice to see that the underlying condition numbers of the operator inversion are taken note of to limit the refinement.