In this chapter, we introduce a wavelet multiscale representation for geometric regions and geometric integrals in Euclidean space. The basic idea is to represent a geometric region in terms of the characteristic function of its interior. By differentiating this function, we obtain a measure supported on the boundary of the region. For instance, for an interval, this measure would consist of Dirac delta measures supported at the endpoints. By using the connection coefficients introduced in the preceding chapter, we are able to formulate a multiresolution expansion for the boundary measure and, consequently, for boundary integrals. These boundary integrals have been used extensively in the wavelet—Galerkin solution for certain types of elliptic boundary value problems as discussed in Chapter 12.
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