Consider the Gaussian function . It is well-known that, if we define the Fourier transform of a Schwartz function to be
then . The proof of this result I learned as an undergraduate used Cauchy’s theorem of complex analysis. I did not really understand the proof at the time, as I was not to take complex analysis until the following semester. However, I would like to describe a (well-known) proof of this result which relies on properties of the Fourier transform with respect to differentiation and global uniqueness of solutions to certain ordinary differential equations (ODEs).
Let be defined as above. Observe that by the Picard-Lindelof theorem, uniquely satisfies the ordinary differential equation
Since is Schwartz function (i.e. ), we can take its Fourier transform to another Schwartz function . Taking the Fourier transform of both sides of the ODE above, we see that satisfies
By uniqueness, we conclude that , for all .
We can use this one-dimensional result to compute the Fourier transform of the -dimensional Gaussian
It follows from the one-dimensional case , and therefore the integral defining makes sense and is absolutely convergent. By Fubini’s theorem,
I’ll later upload to the problem solutions section of this site the complex-analytic proof I learned as an undergrad.