Representation of Signals as Series of Orthogonal Functions

This paper gives an introduction to the theory of orthogonal projection of functions or signals. Several kinds of decomposition are explored: Fourier, Fourier-Legendre, Fourier-Bessel series for 1D signals, and Spherical Harmonic series for 2D signals. We show how physical conditions and/or geometry can guide the choice of the base of functions for the decomposition. The paper is illustrated with several numerical examples.