Alternatives to Dirac Delta?

Question

In one of its lower paragraphs, Wikipedia describes the dirac delta distribution as the limit of a sequence of zero-centered normal distributions.

https://en.wikipedia.org/wiki/Dirac_delta_function

I don't know much about "distributions" per se, but, are there other integrals of sequences of functions that converge to the dirac delta distribution? Or is this sequence unique in some way?

There is a whole section on that in the very same wiki article -.- — Tobsn, Aug 28 '22 at 19:45

score 2 · Answer 1 · answered Aug 28 '22 at 19:55

2

You can try to prove the following yourself: Let $\chi$ be a continuous nonnegative function such that $$ \int_{-\infty}^{\infty} \chi(x)\,dx = 1 $$ and for each $\delta > 0$, let $$ \chi_\delta(x) = \delta^{-1}\chi(\delta^{-1}x). $$ Then for any bounded continuous function $f$ on the real line, $$ \lim_{\delta\rightarrow 0} \int_{-\infty}^{\infty} \chi_\delta(x)f(x)\,dx = f(0). $$

answered Aug 28 '22 at 19:55

Deane

7,582

This doesn't seem to be a statement about sequences of functions. – askquestions2 Aug 28 '22 at 20:35
@askquestions2, let $\delta = \frac{1}{k}$, $k \in \mathbb{Z}^+$. – Deane Aug 28 '22 at 21:05

Alternatives to Dirac Delta?

1 Answers1