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We review some recent results on multiwavelet methods for solving integral and partial differential equations and present an efficient representation of operators using discontinuous multiwavelet bases, including the case for singular integral operators. Numerical calculus using these representations produces fast O(N) methods for multiscale solution of integral equations when combined with low separation rank methods. Using this formulation, we compute the Hilbert transform and solve the Poisson and Schrödinger equations. For a fixed order of multiwavelets and for arbitrary but finite-precision computations, the computational complexity is O(N). The computational structures are similar to fast multipole methods but are more generic in yielding fast O(N) algorithm development.
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