Complex physical phenomena, signals and images involve structures of very different scales. A wavelet transform operates as a zoom, which simplifies the analysis by separating local variations at different scales. Yves Meyer found wavelet orthonormal bases having better properties than Fourier bases to characterize local properties of functions, physical measurements and signals. This discovery created a major scientific catalysis, which regrouped physicists, engineers and mathematicians, leading to a coherent theory of multiscale wavelet decompositions with a multitude of applications.
This lecture will explain the construction of Meyer wavelet bases and their generalization with fast computations. We shall follow the path of this human adventure, with ideas independently developed by scientists working in quantum physics, geophysics, image and signal processing but also neurophysiology of perception. The synthesis in the 1980's provoked by Yves Meyer's work was an encounter between applications and a pure harmonic analysis research program, initiated by Littlewood-Paley in the 1930's. It remains at the roots of open mathematical problems in high-dimension, for physics and big data analysis.