Exceptional sets and wavelet packets orthonormal bases

We give a partial positive answer to a problem posed by Coifman et al. in [1]. Indeed, starting from the transfer function m0 arising from the Meyer wavelet and assuming m_0=1 only on [–\pi/3, \pi/3], we provide an example of pairwise disjoint dyadic intervals of the form I(n, q)=[2qn, 2q(n+1)), (n, q)\in E\subset N×Z, which cover [0, +\infty) except for a set A of Hausdorff dimension equal to 1/2, and such that the corresponding wavelet packets 2^{q/2}w_n (2^qx–k), k\in Z, (n, q)\in E\subsetN×Z form an orthonormal basis of L^2(R).