Description:
Hilbert transform pairs of wavelets, biorthogonal wavelets and frames were found to be attractive in many applications. The Hilbert transform pairs are, however, hardly adoptable for applications, since their two-scale symbols are not trigonometric polynomials. Moreover, the symbols can not be implemented as FIR filters, nor rational IIR filters. That is the reason why approximations are constructed by many researchers in spite of the theoretical existence of the Hilbert transform pairs of wavelets, biorthogonal wavelets, and frames. But these conventional approaches have two drawbacks. Firstly, the wavelets and refinable functions do not have closed forms. Secondly, the symmetry, or "linear phase", of the wavelets and refinable functions is an important constraint in many applications. But, the results, however, show that it is not easy to get symmetric Hilbert transform pairs of wavelets (or generators of frames) ...