The 2021 course by Stéphane Mallat, Professor at the Collège de France, focuses on exploring the triptych of 'Regularity, Approximation, Sparsity.' While the courses since 2018 have focused on deep neural networks, this year is oriented towards signal processing (data). Two classical themes are addressed: approximation in low dimension and denoising/compression. These themes are not far from those of statistical learning because we need to consider sparse representations that exploit the regularities of the signal (the data). Topics reviewed include Fourier analysis, Shannon's theorem, and multiresolution analyses by wavelets with an application to JPEG and JPEG2000 image compression.