Non-orthogonal wavelets
Yves Meyer · Cambridge University Press eBooks · 1993
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 L 2 convergence of Morlet's iterative algorithm. Unlike the case of orthogonal wavelets, L 2 convergence does not necessarily imply that “Morlet's wavelets” can be used in any function space other than the reference space L 2 . 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: (a) the collection of functions 2 j /2 θ(2 j x – κ), j , κ ∈ ℤ, is a Riesz basis of L 2 (ℝ); (b) the above collection is not complete in L p (ℝ). These properties are not due to any special pathology of the spaces L p (ℝ), 2 < p < ∞, which are, in any case, not in the least pathological. The same happens if we try to decompose the Holder spaces C α using non-orthogonal wavelets.