on the eigenfunctions and eigenvalues of the fourier transform 
[Jun. 22nd, 201102:37 pm]
Tobin Fricke's Lab Notebook

Recently my interest was piqued in the eigenfunctions of the Fourier transform. The Fourier transform is a linear operator on a space of functions, so it has eigenvectors and eigenvalues: functions who are their own Fourier transform. I think many people know that "the Fourier transform of a Gaussian is a Gaussian", but the other eigenfunctions are not so well known.
It is relatively easy to show that taking the Fourier transform four times in succession is an identity operation. If you remember that taking the complex conjugate in the frequency domain is equivalent to time reversal in the time domain, this is easy to see. Taking the Fourier transform twice gives you time reversal, so taking it four times gives you the identity.
This means that the eigenvalues of the Fourier transform have to obey x^4 = 1, i.e. the eigenvalues of the Fourier transform are the 4th roots of unity: {1, i, 1, i} (ref) This, of course, agrees with our knowledge that the Fourier transform is unitary. This is Parseval's theorem: the RMS of a function and its transform are equal.
With each eigenvalue we can associate a set of eigenfunctions who have that eigenvalue. Call these sets H0, H1, H2, H3. The Fourier transform of a function in Hn is just that function times an eigenvalue of i^n. We can define the Fourier transform in terms of these sets:
F{f} = f H_{0}⟩⟨H_{0} + i f H_{1}⟩⟨H_{1} i f H_{2}⟩⟨H_{2}  i fH_{3}⟩⟨H_{3}
where H_{n}⟩⟨H_{n} is the projection operator onto the subspace Hn.
It is curious to me that there are only four eigenvalues, and thus these spaces must be very big. There must be very many ways to parametrize each of the families Hnvery many different bases.
It turns out that one such basis is very familiar to physicists: HermiteGauss functions (i.e. a Gaussian multiplied by a Hermite polynomial) are eigenfunctions of the Fourier transform. These show up very often in physics; two particular examples come to mind:
1. the energy eigenstate wavefunctions of the quantum simple harmonic oscillator 2. the HermiteGauss modes of laser resonators
In his book "Fourier Analysis", Javier Duoandikoetxea (what a name!) (page 22, available on the amazon preview, search for "Eigenfunctions") tells us that the HermiteGauss functions provide a complete basis for L^2, i.e. the space of squareintegrable functions:
h_n(x) = ((1)^n / n!) exp( π x^2) (d/dx)^n exp(π x^2)
F{h_n} = (i)^n h_n
Let e_n be a normalized version of h_n:
e_n = h_n /  h_n  = Sqrt[(4π)^(n) Sqrt[2] n!] h_n
Then {e_n} is an orthonormal basis for L^2, and the Fourier transform may be written
F{f} = sum of (i)^n ⟨fe_{n}⟩ over n in ℤ
[Duoandikoetxea says that this is the approach taken by Norbert Wiener in "The Fourier Integral and Certain of its Applications".]
Why do these HermiteGauss functions show up in the physical situations mentioned earlier? If we transform to unitless variables, the Hamiltonian of the harmonic oscillator is simply:
H = x^2 + p^2
This equation is symmetric under interchange of x and p. Finding the energy eigenstates means solving for the eigenstates ψ such that E ψ = H ψ where E is a scalar. Symmetry under interchanging x and p means the coordinatespace and momentumspace representations of the wavefunction must be the same. How do we transform the wavefunction from position space to momentum space? We take the fourier transform. Thus the energy eigenstates of the harmonic oscillator must also be eigenfunctions of the Fourier transform.
But why these HermiteGauss functions in particular? We can take any linear combination of functions all in the same subspace Hn and get an eigenfunction of the Fourier transform, but this won't in general be an energy eigenstate of the SHO. After all, the Fourier transform has only four eigenvalues, but the SHO Hamiltonian has an entire ladder of eigenvalues.
What other properties are needed to uniquely define the HermiteGauss functions? (Related question on math.SE: How do I compute the eigenfunctions of the Fourier Transform?) 

