Decomposition on a Hermite basis
Being the eigenstates of the harmonic oscillator, a set of Hermite functions is
a basis of the Hilbert state. This amazing consequence of Hilbert mathematics is
illustrated here where you can change a target function chosen among a rectangle,
a triangle and a gaussian by changing its center and width.
In fact, the decomposition is exact only when an infinite number of basis
functions is used. When truncating the basis to a finite number of element, the
decomposition is only approximate.