Reverse engineering Fourier conventions

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 2, is +1 or -1, and q is 2 or 1.

I'm posting these notes for my future reference and for anyone else who may need to do the same sleuthing.

Notation

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.

Convolution theorem

The convolution theorem gives a quick way to determine the parameter m. Suppose convolution is defined by

Then

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 2 out front, then m = 2. 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 = 2 and m = 1 or q = 1 and m = 2. 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 = 2, then q = m = 1. If k= 1/2, then q = m = 1.

Double transform

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.

We have

with the same conclusions based on k as above:

• If k = 1, then either q = 2 and m = 1 or q = 1 and m = 2.
• If k = 2, then q = m = 1.
• If k= 1/2, then q = m = 1.

Shift and differentiation

So far we none of our theorems have allowed us to reverse engineer . Either the shift or differentiation theorem will let is find q.

The shift theorem says

where k = q. Since = 1 and q = 1 or 2, the product q determines both and q.

Similarly, the differentiation theorem says the Fourier transform of the derivative of f transforms as

and again k = q.

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