Wavelet diagonalization of convolution operators
Fredrik Ekstedt · 1997
It is well known that wavelets cannot be eigenfunctions of differential operators. We show that for homogeneous convolution operators, one can obtain a diagonal representation using two different biorthogonal wavelet bases, properly adapted to the operator at hand. We generalize this to include many inhomogeneous convolution operators, using "wavelet-like" basis functions, i.e. functions that share all the important properties of classical wavelets but not necessarily are dilates and translates of a single mother wavelet. We also show how to associate a multiresolution structure to these bases, which means that the wavelet transforms involved can be implemented with fast algorithms. Finally, we use these techniques to construct a fast wavelet transform for complex-valued signals that separates positive and negative frequencies, which is important in the analysis of radar signals.