We study the uncertainty principles of Hardy and of Beurling, and functions that "only just" satisfy the inequalities of uncertainty principles. More specifically, we show that if a function and its Fourier transform have nearly gaussian decay, the the coefficients of its Hermite expansion decay fast, and vice versa. We give a new and simple proof of generalisation of Beurling's uncertainty principle first in R using complex analysis. Then we generalise to R^n using various techniques. Also we illustrate connections with the classical moment problem.