The Gaussians as good kernels

Gaussians

For $f$ a function on $\mathbb{R}$, we call $f$ a Gaussian if for $a>0$, $f=e^{-\pi a{x}^2}$

Proposition

We have ${\int }_{-\infty }^{\infty }{e}^{-\pi x^2}\text{d}x=1$

Proof To see why this is true, we use the multiplicative property of the exponential to reduce the calculation to a two-dimensional integral. More precisely, we can argue as follows:

$\begin{array}{rl}{\left({\int }_{-\infty }^{\infty }{e}^{-\pi x^2}dx\right)}^2& ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{-\pi (x^2+y^2)}dxdy\\ & ={\int }_0^{2\pi }{\int }_0^{\infty }{e}^{-\pi r^2}rdrd\theta \\ & ={\int }_0^{\infty }2\pi r{e}^{-\pi r^2}dr\\ & ={\left[-e^{-\pi r^2}\right]}_0^{\infty }\\ & =1,\end{array}$ where we have evaluated the two-dimensional integral using polar coordinates.

Theorem

If $f(x)=e^{-\pi x^2}$, then $\hat{f}(\xi )=f(\xi )$.

Proof. Define $F(\xi )=\hat{f}(\xi )={\int }_{-\infty }^{\infty }{e}^{-\pi x^2}{e}^{-2\pi ix\xi }dx,$ and observe that $F(0)=1$, by our previous calculation. By property (v) in proposition , and the fact that $f^{\prime }(x)=-2\pi xf(x)$, we obtain $F^{\prime }(\xi )={\int }_{-\infty }^{\infty }f(x)(-2\pi ix)e^{-2\pi ix\xi }dx=i{\int }_{-\infty }^{\infty }{f}^{\prime }(x)e^{-2\pi ix\xi }dx.$ By (iv) of the same proposition, we find that $F^{\prime }(\xi )=i(2\pi i\xi )\hat{f}(\xi )=-2\pi \xi F(\xi ).$ If we define $G(\xi )=F(\xi )e^{\pi {\xi }^2}$, then from what we have seen above, it follows that $G^{\prime }(\xi )=0$, hence $G$ is constant. Since $F(0)=1$, we conclude that $G$ is identically equal to $1$, therefore $F(\xi )=e^{-\pi {\xi }^2}$, as was to be shown.

Corollary

If $\delta >0$, and $K_{\delta }(x)={\delta }^{-\frac{1}{2}}{e}^{-\frac{\pi x^2}{\delta }}$, then $\widehat{K_{\delta }}(\xi )=e^{-\pi \delta {\xi }^2}$

We directly get this through scaling property.

Remark

In this case we can see $K_{\delta }$ and ${\hat{K}}_{\delta }$ cannot both be localized (that is, concentrated) at the origin. This is an example of a general phenomenon called the Heisenberg uncertainty principle.

Proposition

With $K_{\delta }(x)={\delta }^{-\frac{1}{2}}{e}^{-\frac{\pi x^2}{\delta }}$, we have (i) ${\int }_{-\infty }^{\infty }{K}_{\delta }(x)\text{d}x=1$ (ii) ${\int }_{-\infty }^{\infty }|K_{\delta }(x)|\text{d}x\le M$ (iii) for every $\beta >0$, we have ${\int }_{|x|>\beta }|K_{\delta }(x)|\text{d}x\to 0$ as $\delta \to 0$

Theorem

The collection $\{K_{\delta }{\}}_{\delta >0}$ is a family of good kernels as $\delta \to 0$.

Corollary

If $f\in \mathcal{S}(\mathbb{R})$, then $(f\star K_{\delta }(x))\to f(x)\text{uniformly in x as}\delta \to 0.$