In this question, everything happens in $\mathbb{R}$.

I've seen the definition of the weak derivative as an operator on the space of distributions, which makes $$(D\Lambda)(\phi):=-\Lambda(\phi')$$ for $\phi$ any smooth function with compact support. From this definition, a straightforward computation using the fundamental theorem of calculus shows that the weak derivative of the Heaviside function is the Dirac delta ($\delta(\phi)=\phi(0)$).

I am trying to see how this situation happens in the Sobolev space $H^s$ defined as $\ell^2_s(\mathbb{Z})$ with counting measure weighted by $(1+z^2)^s$, but I'm a bit lost.

I think that, due to Parseval's theorem, we should expect to associate the action $\Lambda(\phi)$ of a distribution on a smooth function with compact support to $\langle u,\phi \rangle_2$, where the inner product is just the inner product of $\ell^2(\mathbb{Z})$, and $\Lambda$ is associated to $u$. But maybe since in this second definition we come from $L^2(S^1) \sim L^2(0,2\pi)$, we should expect to act only on $\phi$ smooth $2\pi$-periodic functions, not compactly supported ones. Anyway, I end up with the action of the weak derivative of the Heaviside on a function being a series over the odd Fourier coefficients of the function, which I don't know how to relate to the value of the function on the point where the break of the Heaviside occurs.

My question is: am I on the wrong path in this? If so, how should I reconcile the Dirac delta in both definitions?

  • 2
    $\begingroup$ You cannot define the Dirac delta as the derivative of a function on the circle - $df$ always integrates to zero. So you can't expect to write the dirac delta itself as the derivative of a periodic function. You could write a characteristic function for some closed interval and take its derivative to get a two-poled Dirac delta. In any case, the Dirac delta is in $L^2_{-1}$, not $L^2$. $\endgroup$ – user98602 Mar 20 '17 at 6:34
  • $\begingroup$ @MikeMiller I know, sorry if my question is not clear. I am working through the Sobolev spaces part on Warner (differential topology and Lie groups). What I did is take $\mathbf{1}_{(\pi,2\pi]}$ and compute its Fourier coefficients, then apply the "weak derivative operator" on the coefficients. However, what I get is a bundle of coefficients which I do not know how to interpret as the Diract delta. Is my problem clearer? $\endgroup$ – Aloizio Macedo Mar 20 '17 at 23:10

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