Dirac delta function as the limit of a seqquence

[krishna mohan]

Dirac delta function as the limit of a sequence

Hi..

If I have a sequence which in some limit tends to infinity for x=0 and goes to zero for x$$\neq$$0, then can I call the limit as a dirac delta function?

If not, what are the additional constraints to be satisfied?

 

[krishna mohan]

Let me refine that..

If the integral of a member of the sequence over the whole real line is one, along with the conditions stated in previous post, can the limit of the sequence be identified with a dirac delta?

Or are the conditions of the first post enough??

 

[Hurkyl]

Let f_n be any sequence of positive functions that converges (in the sense of distributions) to the dirac delta.

Then 2f_n does not converge to the dirac delta.

Also, I believe $\sqrt{f_n}$ converges to 0.

 

[Hurkyl]

Also,

$$\lim_{n \rightarrow +\infty} g_n(x) = 0$$​

does not imply

$$\lim_{n \rightarrow +\infty} g_n = 0$$​

(the second limit in the sense of distributions)

In fact, I'm pretty sure you can find a sequence that satisfies the first equation whose limit is $\delta(x-1)$.