Construction of Hilbert transform pairs of wavelet bases and optimal timefrequency localization

We propose a novel method of constructing exact Hilbert transform (HT) pairs of wavelet bases using fractional B-splines and state necessary and sufficient conditions for gen-erating such wavelet pairs. In particular, we demonstrate how HT pairs of biorthogonal wavelet bases of L2(R) can be con-structed using well-localized scaling functions with identical Riesz bounds. Finally, we illustrate this concept by construct-ing a family of analytic Gabor-like wavelets that exhibit near optimal time-frequency localization