Microlocal filtering with multiwavelet frames

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Our orthonormal multiwavelet bases, which can decompose functions in the Hilbert space L^2(R^n) microlocally, are shown to be a "stepwise" unconditional basis in L^p(R^n) (1<p<∞) and other related spaces. As part of the proof, an elementary proof of the L^p(R^n) version of the sampling theorem with unconditional convergence is given. Finally, an application is given to the expression of some distributions as sums of boundary values of holomorphic functions.Orthogonal multiwavelets, whose Fourier transforms consist of characteristic functions of squares or sectors of annuli, are constructed in the Fourier domain and are shown to satisfy a multiresolution analysis with several choices of scaling functions. Redundant smooth tight wavelet frames are obtained and these nonorthogonal frame wavelets can be generated by two-scale equations from, a multiresolution analysis. Singularities can be localized in position and direction and the original images can be restored from the scarred images.