Harmonic Analysis

The origins of Harmonic Analysis lie in the work of Euler [Eu] and the Bernoullis who proposed to write periodic functions in terms of the exponentials e inx , n ∈ ℤ, in their study of the vibrating string. It is known that every C∞-function which is 2π-periodic in the real line has an expansion of the form En $$\sum\nolimits_{n = - \infty }^{ + \infty } {{a_n}{e^{inx}}} $$ (we remind the reader one can estimate these coefficients a n very precisely, and that we do not need to restrict ourselves to C∞-functions). It was the work of Fourier [Fo] on heat conduction that showed, once and for all, the importance and the interest of such expansions, and since then they have been called Fourier expansions. It is clear that another way of saying that a function f is periodic with period τ is to say that f satisfies the convolution equation $$\left( {{\delta _\tau } - \delta } \right) * f = 0$$