ODE Proof of Gaussian Fourier Transform

Consider the Gaussian function f(x) := e^{-\pi x^{2}}. It is well-known that, if we define the Fourier transform of a Schwartz function g: \mathbb{R}^{n} \rightarrow \mathbb{R} to be

\displaystyle \widehat{g}(\xi) := \int_{\mathbb{R}^{n}}g(x)e^{-2\pi i x\cdot\xi}dx, \indent \forall \xi \in \mathbb{R}^{n}

then \widehat{f}(\xi) = f(\xi). 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 f be defined as above. Observe that by the Picard-Lindelof theorem, f uniquely satisfies the ordinary differential equation

\displaystyle (\partial_{x}f)(x) = -2\pi x f(x), \indent \forall x \in \mathbb{R}

Since f is Schwartz function (i.e. f \in \mathcal{S}(\mathbb{R})), we can take its Fourier transform to another Schwartz function \widehat{f}. Taking the Fourier transform of both sides of the ODE above, we see that \widehat{f} satisfies

\displaystyle (2\pi i\xi) \widehat{f}(\xi) = -i(\partial_{x}\widehat{f})(\xi) \Longleftrightarrow -2\pi\xi\widehat{f}(\xi) = (\partial_{x}\widehat{f})(\xi), \indent \forall \xi \in \mathbb{R}

By uniqueness, we conclude that \widehat{f}(\xi) = e^{-\pi\xi^{2}}, for all \xi \in \mathbb{R}.

We can use this one-dimensional result to compute the Fourier transform of the n-dimensional Gaussian

\displaystyle f(x) = f(x_{1},\cdots,x_{n}) = e^{-\pi\left|x\right|^{2}} = e^{-\pi(x_{1}^{2}+\cdots+x_{n}^{2})}, \indent \forall x \in \mathbb{R}^{n}

It follows from the one-dimensional case f \in \mathcal{S}(\mathbb{R}^{n}), and therefore the integral defining \widehat{f} makes sense and is absolutely convergent. By Fubini’s theorem,

\displaystyle \begin{array}{lcl}\widehat{f}(\xi) = \int_{\mathbb{R}^{n}}e^{-\pi\left|x\right|^{2}}e^{-2\pi i \xi \cdot x}dx &=& \left(\int_{\mathbb{R}}e^{-\pi x_{1}^{2}}e^{-2\pi i \xi_{1}x_{1}}dx_{1}\right)\cdots\left(\int_{\mathbb{R}}e^{-\pi x_{n}^{2}}e^{-2\pi i \xi_{n}x_{n}}dx_{n}\right)\\&=& e^{-\pi \xi_{1}^{2}}\cdots e^{-\pi\xi_{n}^{2}}\\&=& e^{-\pi\left|\xi\right|^{2}}\end{array}

I’ll later upload to the problem solutions section of this site the complex-analytic proof I learned as an undergrad.

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One Response to ODE Proof of Gaussian Fourier Transform

  1. Pingback: Complex-Analytic Proof of Fourier Transform of Gaussian | Math by Matt

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