The most annoying thing about Fourier analysis is that there are many slightly different definitions of the Fourier transform. One time I got sufficiently annoyed that I creates a sort of Rosetta Stone of Fourier theorems under eight different conventions. Later I discovered that Mathematica supports these same eight definitions, but with slightly different notation.
When I created my Rosetta Stone I wanted to have a set of notes that answered the question “What are the basic Fourier theorems under this convention?” Recently I was reading a reference and wanted to answer the opposite question “Given the theorems this book is stating, what convention must they be using?”
The eight definitions correspond to
where m is either 1 or 2π, σ is +1 or -1, and q is 2π or 1.
I’m posting these notes for my future reference and for anyone else who may need to do the same sleuthing.
For the rest of the post, let F and G be the Fourier transforms of f and g respectively. We write
for the pair of a function and its Fourier transform.
Define the inner product of f and g as
if f and g are real-value. If the functions are complex-valued, replace g with the complex conjugate if g.
We will sometimes denote the Fourier transform of a function by putting a hat on top of it.
The convolution theorem gives a quick way to determine the parameter m. Suppose convolution is defined by
and so you can find m immediately. If f*g = F*G with no extra factor out front, m = 1. Otherwise if there’s a factor of √2π out front, then m = 2π. If there’s any other factor, you’ve got an arcane definition of Fourier transform that isn’t one of the eight considered here.
Some authors, like Walter Rudin, include a scaling factor in the definition of convolution, in which casethe argument of this section doesn’t hold.
Parseval and Plancherel
Parseval’s theorem says that the inner product of f and g is proportional to the inner product of F and G. The proportionality constant depends on the definition of Fourier transform, specifically on m and q, and so you can determine m or q from the form of Parseval’s theorem.
If k = 1, then either q = 2π and m = 1 or q = 1 and m = 2π. If you know m, say from the statement of the convolution theorem, then Parseval’s theorem tells you q.
Plancherel’s theorem is the special case of Parseval’s theorem with f = g. It can be used the same way to solve for m or q.
If k = 2π, then q = m = 1. If k= 1/2π, then q = m = 1.
Theorems about taking the Fourier transform twice carry the same information as Parseval’s and Plancherel’s theorems, i.e. they also let you determine m or q.
with the same conclusions based on k as above:
- If k = 1, then either q = 2π and m = 1 or q = 1 and m = 2π.
- If k = 2π, then q = m = 1.
- If k= 1/2π, then q = m = 1.
Shift and differentiation
So far we none of our theorems have allowed us to reverse engineer σ. Either the shift or differentiation theorem will let is find σq.
The shift theorem says
where k = σq. Since σ = ±1 and q = 1 or 2π, the product σq determines both σ and q.
Similarly, the differentiation theorem says the Fourier transform of the derivative of f transforms as
and again k = σq.