Introduction

Well before orthonormal wavelet bases existed, wavelets had been used by J. Morlet (a geophysical engineer with O.R.I.C, Elf-Aquitaine) for the numerical processing of seismic signals recorded during oil prospecting expeditions.

Morlet's methods were mathematically justified, post facto, by Daubechies ([87]) and this chapter is dedicated to the statement and proof of the L2 convergence of Morlet's iterative algorithm.

Unlike the case of orthogonal wavelets, L2 convergence does not necessarily imply that “Morlet's wavelets” can be used in any function space other than the reference space L2. In fact, results by P. Tchamitchian and then by P.G. Lemarié have enabled the following to be established: for every exponent p > 2, there exists a function θ(x), of the real variable x, belonging to the Schwartz class S(ℝ), all of whose moments are zero and which satisfies two apparently contradictory properties as follows:

1. (a) the collection of functions 2j/2θ(2jx – κ), j, κ ∈ ℤ, is a Riesz basis of L2(ℝ);

2. (b) the above collection is not complete in Lp(ℝ).

These properties are not due to any special pathology of the spaces Lp(ℝ), 2 < p < ∞, which are, in any case, not in the least pathological. The same happens if we try to decompose the Holder spaces Cα using non-orthogonal wavelets.