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 m0=1 only on [−π/3, π/3], we provide an example of pairwise disjoint dyadic intervals of the form I(n, q)=[2qn, 2q(n+1)), (n, q)εE⊂N×Z, which cover [0, +∞) except for a set A of Hausdorff dimension equal to 1/2, and such that the corresponding wavelet packets 2q/2wn (2qx−k), kεZ, (n, q)εE⊂N×Z form an orthonormal basis of L2(R).