Dirac delta approximation - need an outline of a simple and routine proof

[hooker27]

Hi, I need your help with a very standard proof, I'll be happy if you give me some detailed outline - the necessary steps I must follow with some extra clues so that I'm not lost the moment I start - and I'll hopefully finish it myself. I am disappointed that I can't proof this all by myself, but I really need to move on with my work so any help here will be appreciated.

Let $$\theta$$ be a function that decays faster than any polynomial, is smooth or whatever else is required. Let

$$\int_{-\infty}^{\infty} \theta = K$$

I want to show, that for all reasonable $$f$$ (continuous, smooth, bounded, ... again - whatever is required) the following identity holds

$$\lim_{a \rightarrow 0}\int_{-\infty}^{+\infty} f(x) \frac{1}{a}\theta\left(\frac{x}{a}\right) dx = K f(0)$$

In other words - the limit

$$\lim_{a \rightarrow 0} \frac{1}{a}\theta\left(\frac{x}{a}\right)$$

is a multiple of Dirac delta, in the sense of distributions. The proof needs not to be 100% precise, even 80% precise, I need it for some little physics paper I write and physicists are seldom 100% precise (in math, anyway), but I need something a little more rigorous than "Let's assume $$\theta$$ is rectengular..." for which the proof is way too easy.

I appreciate any help, thanks in advance. This is no hw, if it's any important.

 

[Hurkyl]

What lines of attack have you tried on this? It's hard to give helpful hints without knowing where you're at on the problem!

Anyways, two high-level observations that are almost surely useful to the proof (or at least to the devising of a line of attack) are

$$\int_{-\infty}^{+\infty} \theta \approx \int_{-H}^{+H} \theta$$

and

[tex]f(\epsilon) \approx f(0),[/itex]

capturing the notions that $\theta$ is rapidly decaying and that f is continuous at 0, respectively.

(H is a 'large' positive number, $\epsilon$ is a 'small' positive number)


(I use the integral expressing that the tails are irrelevant, rather than something more direct like $\theta(\pm H) \approx 0$, because the integral actually appears in the data you've given)

 

[maze]

Are you sure you don't want $\frac{1}{a}\theta (a x)$?