5. Generalized wavelets

The problem of adapting multiscale representations and wavelets to bounded domains $\mathrm{\Omega }\subset {ℝ}^d$ is relatively straightforward in signal and image processing, since the domain is usually an interval or a square. This problem becomes considerably more complicated in situations encountered in numerical simulation, where the domain of definition of the functions may have a more general geometry, which does not lend itself to the discretizations on Cartesian grids underlying tensor product multiscale decompositions. We may also wish to incorporate edge conditions on these representations, for example in the context of the numerical solution of a partial differential equation.

Since...