0 item(s) in cart

Shopping Cart
We accept VISA Mastercard American Express Discover

Properties of the Fourier Transform Operator

Fourier Transform Operations

Many people are familiar with the famous bell shaped curve of statistics. (It is the result of the “central limit theorem” – and beloved by quants who managed to crash the economy by ignoring the caveats in its derivation. It is also the solution of the harmonic oscillator in quantum mechanics.) It has always seemed interesting that the Fourier transform of the Gaussian is another Gaussian:

Here I have chosen the Gaussians so that the one on the left is the same as the one on the right. (If the Gaussian on the right is made narrower, the one on the left is wider etc.)

I choose to normalize the Fourier transform with equation for reasons of simplification of the equations below.

We (that is myself and whoever is reading this) define the Fourier transform of a function f as follows:

equation

We can write this in the compressed form:

g=Ff

Here F represents (1/ thought of as an operator, and the product of F and f implies an integral in analogy with matrix multiplication (wherein the product implies a sum).

The inverse Fourier transform is:

equation

And we can write the inverse Fourier operator as

Fourier Transform Operations

equation

(Now we see why the normalization is equation . )

The inverse transform can be simply written:

equation


Now to return to the subject of the Gaussian, in the language of operator algebra it is an eigenfunction of the Fourier transform and in fact of eigenvalue 1. We are led to ask what might other eigenfunctions of F be, and what are their eigenvalues:

equation

The product of two Fourier transforms leads to a delta function:

equation

This operator just replaces t with -s in the argument of a function to which it is applied.

Finally  is just the inverse transform and  is the identity operator:

equation

equation

This last expression implies that the eigenvalues of the Fourier transform are all forth roots of one, that is: 1, i,-1,-i.

Just as we can find the even and odd parts of any function we can find the parts of any function which are eigenfunctions with the four eigenvalues:

equation

equation

equation

equation

And therefore:

equation

So if we want to find a function which is an eigenvector of the Fourier transform we may apply any of the above to derive an eigenfunction. Unfortunately most of these functions look pretty silly: Take for example the square pulse,

equation

Its Fourier transform is

equation

Fourier Transform Operations

Therefore the eigenfunction equation is just ½ the sum of the pulse and its transform, certainly nothing as elegant as the Gaussian.

Here are some Fourier eigenfunctions which have eigenvalue 1 and are (in my opinion) rather nice:

equation

equation

And most amazingly with eigenvalues equation (Where equation are the Hermite polynomials.):

equation