Origin of $\int_{-\infty}^{+\infty}{\delta(t)x(t) dt}=x(0)$

Question

Where $\int_{-\infty}^{+\infty}{\delta(t)x(t) dt}=x(0)$ comes from? Maybe this derives from the fact that there are families of sequences $f_n(t)$ of functions (e.g. rectangular functions, gaussian functions etc.) that, for $n\rightarrow+\infty$, gives $\delta(t)$?

For the above sequences exist the following integral:

$$\mid\int_{-\infty}^{+\infty}{f_n(t)x(t) dt}-x(0)\mid<\epsilon$$

which proof that the above sequences are approximations of the Dirac delta.

Thank you in advance.

Answer 1

• The origin is the Fourier series theorem : let the Dirichlet kernel $$ D_N(t) = \sum_{n=-N}^N e^{int} = \frac{\sin\left(\left(N +1/2\right) t \right)}{\sin(t/2)}$$ then $$\lim_{N \to \infty} \int_{-\pi}^\pi D_N(t) \phi(t) dt = 2\pi\phi(0)$$ whenever $\phi$ is $C^1$. As $D_N$ is $2\pi$-periodic then $\lim_{N \to \infty} \int_{-\infty}^\infty D_N(t) \phi(t) dt = 2\pi\sum_n\phi(2\pi n)$ whenever $\phi$ is $C^1$ and compactly supported, ie. $D_N \to 2\pi\sum_n \delta(t-2\pi n)$ in the sense of distributions

• This generalizes to the Fourier inversion theorem : let $$K_N(t) =\int_{-N}^N e^{i \omega t} d\omega= \frac{2\sin(N t)}{ t}$$ then $$\lim_{N \to \infty} \int_{-\infty}^\infty K_N(t) \phi(t) dt = 2\pi\phi(0)$$ whenever $\phi$ is $C^1$ and fast decreasing, ie. $K_N \to 2\pi\delta$ is the sense of distributions.

Answer 2

This comes from the writings of Paul Dirac. He was a mathematical physicist, and wrote things that don't make sense mathematically. But (the reasoning goes) if the results correctly predict some physics, then the calculations are permitted. This differs from mathematicians, who say that the calculations are permitted only if there is a mathematical proof.