Wavelets on Closed Subsets of the Real Line

We construct orthogonal and biorthogonal wavelets on a given closed subset of the real line. We also study wavelets satisfying certain types of boundary conditions. We introduce the concept of \wavelet prob-ing", which is closely related to our construction of wavelets. This technique allows us to very quickly perform a number of diierent numerical tasks associated with wavelets. x1. Introduction Wavelets and multiscale analysis have emerged in a number of diierent elds, from harmonic analysis and partial diierential equations in pure mathematics to signal and image processing in computer science and electrical engineering. Typically a general function, signal, or image is broken up into linear combinations of translated and scaled versions of some simple, basic building blocks. Multiscale analysis comes with a natural hierarchical structure obtained by only considering the linear combinations of building blocks up to a certain scale. This hierarchical structure is particularly suited for fast numerical implementations; the underlying idea being that functions on a certain scale only need to be sampled at a rate approximately given by the scale they live on. To discuss this in more concrete terms, let us consider a standard, orthogonal wavelet decomposition of a general function f on the line. This is a representation in terms of linear combinations of translated dilates of a single function : f(x) =