Measures associated to wavelet packets

We study the properties of the continuous measures m_k , induced by the wavelet packet algorithm on the Borel sets of [01), in the case of Lemarié–Meyer wavelet. It is still an open problem to determine if these measures are absolutely continuous with respect to the Lebesgue measure. This problem was formulated by Coifman, Meyer and Wickerhauser in [2]. In order to understand if these measures are absolutely continuous or not, it is important to know their Fourier coefficients. We achieve this goal in two steps. First we provide explicit formulas for the values of m_k in dyadic intervals in terms of the wavelet packets, then we show that each m_k is the weak limit of certain probability measures whose Fourier coefficients are easy to calculate.