Implementation of Operators via Filter Banks: Hardy Wavelets and Autocorrelation Shell

Abstract We consider implementation of operators via filter banks in the framework of the multiresolution analysis. Our method is particularly efficient for convolution operators. Although our method of applying operators to functions may be used with any wavelet basis with a sufficient number of vanishing moments, we distinguish two particular settings, namely, orthogonal bases and the autocorrelation shell. We apply our method to evaluate the Hilbert transform of signals and derive a fast algorithm capable of achieving any given accuracy. We consider the case where the wavelet is the autocorrelation function of another wavelet associated with an orthonormal basis and where our method provides a fast algorithm for the computation of the modulus and the phase of signals. Moreover, the resulting wavelet may be viewed as being (approximately, but with any given accuracy) in the Hardy spaceH2(open face R).