Lagmental Vicfred

The Gaussian Is an Eigenfunction of the Fourier Transform by Vicfred

tags: mathematics, Fourier analysis, convolution, Sobolev spaces

With the 2 pi convention, exp minus pi x squared transforms into itself. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Start locally

Fourier analysis expands a function into frequencies. On the circle the coefficients are \(\widehat f(n)\); on \(\mathbf R\) the transform \(\widehat f(\xi)\) is an integral against an oscillatory exponential.

$$ f(x)=e^{-\pi x^2} $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \widehat f(\xi)=e^{-\pi\xi^2} $$

Compute before generalising

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \int_{\mathbf R}e^{-\pi x^2}e^{-2\pi ix\xi}\,dx=e^{-\pi\xi^2},\qquad\int_{\mathbf R}e^{-\pi x^2}\,dx=1 $$

The global view

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f(x)=e^{-\pi x^2},\\[5pt] \mathsf{C}\;&:\quad \widehat f(\xi)=e^{-\pi\xi^2}. \end{aligned} $$

Edge conditions

Normalization conventions move factors of \(2\pi\) between the transform, inverse transform, derivative rule, and Gaussian formula.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \widehat f(\xi)=e^{-\pi\xi^2} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Mon 07 September 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.