Abstract:

We say that two functions form a Hilbert pair if one of them is the Hilbert transform of the other. Complex curves with real and imaginary parts forming Hilbert pairs have many uses in signal processing, and in particular some clever algorithms call for curves where the real and imaginary parts are a Hilbert pair of real-valued wavelets. In this presentation we will derive a characterization of "Hilbert Pairs of Wavelets". We will explain why it is impossible to design bases of wavelets with this property, and show the approach of Selesnick, Baraniuck and Kingsbury around the obstacles. If time permits, we will also see how to use techniques from Computational Commutative Algebra to design "almost Hilbert" pairs of wavelets with compact support.