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The recently introduced compactly supported wavelets of Daubechies have proven to be very useful in various aspects of signal processing, notably in image compression. The fundamental idea in the application to image compression was to take a digitized image and to use wavelets to provide a multiscale representation of it, and to then discard some information at some scales, and leave information more intact at other scales. The Daubechies wavelets have differentiability properties in addition to their compact support and orthogonality properties. A number of authors have used wavelet system for solving various problems in differential and integral equations. An additional point of view for approaching boundary value problems in partial differential equations were introduced, in which the boundary, the boundary data, and the unknown solution (of a boundary value problem) on the interior of a domain are all uniformly represented in terms of compactly supported wavelet functions in an extrinsic ambient Euclidean space. In this paper we describe multiscale representations of domains and their boundaries and obtain multiscale representations of some of the basic elements of geometric calculus (line integrals, surface measures, etc.) which are then in turn useful for specific numerical calculations in problems in approximate solutions of differential equations. This is similar to the spectral method in solving differential equations (essentially using an orthonormal expansion), but here we use the localization property of wavelets to extend this orthonormal representation to boundary data and geometric boundaries. In this paper we want to indicate how to carry this out.
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