Georgia Institute of Technology College of Sciences

A fundamental result in Harmonic Analysis states that many functions defined over the interval [-\pi,\pi] can be decomposed into a Fourier series, that is, decomposed as sums of sines and cosines with integer frequencies. This allows one to describe very complicated functions in a simple way, and therefore provides with a strong tool to study the properties of different families of functions.However, the above decomposition does not hold -- or holds but is not efficient enough-- if the functions are no longer defined over an interval,( e.g. if a function is defined over a union of two disjoint intervals). We will discuss the question of whether similar decompositions are possible also in such cases, with the frequencies of the sines and cosines possibly being no longer integers.