The Solution of a Problem of Coifman, Meyer, and Wickerhauser on Wavelet Packets.

Wavelet packets provide an algorithm with many applications in signal processing together with a large class of orthonormal bases of L^2(R), each one corresponding to a different splitting of L^2(R) into a direct sum of its closed subspaces. The definition of wavelet packets is due to the work of Coifman, Meyer, and Wickerhauser, as a generalization of the Walsh system. A question has been posed since then: one asks if a (general) wavelet packet system can be an orthonormal basis for L2(R) whenever a certain set linked to the system, called the “exceptional set” has zero Lebesgue measure. This question is reflected in the quality of wavelet packet approximation. In this paper we show that the answer to this question is negative by providing an explicit example. In the proof we make use of the “local trace function” by Dutkay and the generalized shift-invariant system machinery developed by Ron and Shen.