Hi, I need your help with a very standard proof, I'll be happy if you give me some detailed outline - the neccessary 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.(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]\int_{-\infty}^{\infty} \theta = K[/tex]

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

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

In other words - the limit

[tex]\lim_{a \rightarrow 0} \frac{1}{a}\theta\left(\frac{x}{a}\right)[/tex]

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 [tex]\theta[/tex] 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.

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

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