We discuss the basic concepts of waves, including phase velocity, dispersion, group velocity. We show how to use the Fourier principle to construct any general wave from the harmonic waves.

This chapter provides an introduction to uncertainty relations underlying sparse signal recovery. We start with the seminal work by Donoho and Stark (1989), which defines uncertainty relations as upper bounds on the operator norm of the band-limitation operator followed by the time-limitation operator, generalize this theory to arbitrary pairs of operators, and then develop, out of this generalization, the coherence-based uncertainty relations due to Elad and Bruckstein (2002), plus uncertainty relations in terms of concentration of the 1-norm or 2-norm. The theory is completed with set-theoretic uncertainty relations which lead to best possible recovery thresholds in terms of a general measure of parsimony, the Minkowski dimension. We also elaborate on the remarkable connection between uncertainty relations and the “large sieve,” a family of inequalities developed in analytic number theory. We show how uncertainty relations allow one to establish fundamental limits of practical signal recovery problems such as inpainting, declipping, super-resolution, and denoising of signals corrupted by impulse noise or narrowband interference.