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Let $\epsilon>0$ and let $f: (-\epsilon, \epsilon) \to \mathbb{R}$ be a differentiable function. Consider the limit

$$\lim_{x\to 0} \frac{\delta_{f(x)}-\delta_{f(0)}}{x},$$ where $\delta_{f(x)}$ and $\delta_{f(0)}$ are the Dirac measures at $f(x)$ and $f(0)$ respectively. Does this limit converges? The topology that I am considering is with respect to the total variation norm. The problem is that if it does converges I don't have a candidate for it. In fact, if I look $\delta_{f(x)}$ as a distribution, then the limit converges to $f'(0)\delta_{f(0)}'$, however, $\delta_{f(0)}'$ is not a measure.

Any help is appreciated.

Thank you!

EDIT:

I am looking at $\delta_a$ as a (complex) measure. I know that the limit converges if I look $\delta_a$ as a distribution, but I just mentioned that as a fact that can give me some information about the limit, however, the limit is not a measure as observed. Since the space of complex measures $C$ with the total variation norm is a Banach space, I was thinking that since it converges as distribution then I had some hope to converge in $C$. If that is not the case, is there an easy way to see that it doesn't converge?

Math Guy
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1 Answers1

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The dirac delta function is a distribution, to compute its derivative, you have to use $\delta'_a(f)=-\int f'\delta$ where $f$ is a test function which vanishes at the infinity. So you obtain that $\delta'_a(f)=-f'(0)$

Tsemo Aristide
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  • I don't think you answer to the problem. $f$ is not a test function. Math Guy wants to derivate $\delta(f(x))$ (classical abuse of notation). For example starting from an identity which goes back to Dirac : $\delta(x^2-a^2)=\frac{1}{2}a^{-1}(\delta(x-a)+\delta(x+a))$, the derivative would be $\frac{1}{2}a^{-1}(\delta'(x-a)+\delta'(x+a))$. Have a look at the delicious document where I found back this formula. – Jean Marie Aug 10 '16 at 02:25
  • I was thinking according to the title that he wants to compute the derivative of the dirac delta function which can be computed by using the classic definition of the derivative of a distribution – Tsemo Aristide Aug 10 '16 at 02:27
  • Yes, the title doesn't reflect what the OP desires. I have indicated him a good reference I just found about this kind of (delicate on the point of view of rigor) things in StackExchange. – Jean Marie Aug 10 '16 at 02:36
  • I am thinking the limit in the space of complex measures with the total variation norm. By the way, I cannot see the link @JeanMarie – Math Guy Aug 10 '16 at 12:53
  • I cannot see the link either. – YoTengoUnLCD Aug 10 '16 at 13:28