In my analysis class, we are now covering distribution theory. We take the Schwartz space of functions $S(\mathbb{R})$ and its continuous dual space, $S'(\mathbb{R})$, the space of tempered distributions. For $T \in S'(\mathbb{R})$ and $\phi \in S(\mathbb{R})$, we use the notation
$$\langle T,\phi \rangle = T(\phi)$$
and when $T(x)$ is a function, we define $$\langle T,\phi \rangle = \int_{-\infty}^{\infty} T(x)\phi(x)dx$$ For the Foruier transform on tempered distributions, we define $$ \langle \mathcal{F}T,\phi \rangle = \langle T,\mathcal{F}\phi \rangle $$ We are asked to compute the Fourier transform of the following functions as distributions
- $H(x)=\chi_{[0,\infty)} (x)$, the Heaviside step function.
- $xH(x)$.
- $x^2 \sin(x)$
Our textbook says that, in the sense of distributions, one has the derivative $H'(x)=\delta_0(x)$ so I thought I could take the Fourier transform on both sides and get, using the derivative identity $ip \hat{H}=\frac{1}{\sqrt{2\pi}}$ but I do not know if I can "divide" with distributions, given other answers on this site and elsewhere, when dividing we need to take a constant multiple of $\delta$ which I do not know why, and I do not how to compute the constant. For the second part, I thought I could do the derivative and get using the product rule (which I think holds for distributions) $(xH)'=H+xH'=H+x\delta_0$ but I do not know how to compute the Fourier transform of $x\delta_0$. I have managed part 3. I would therefore appreciate any help on 1 and 2, and I thank all helpers.
************* Progress: I have managed part 3 but parts 1 and 2 are still a mystery to me. I do not know how to "divide" in the sense of distributions. I thank all helpers.
Best Answer
I'm using $$ \mathcal{F}\{f(x)\} = \int_{-\infty}^{\infty} f(x)e^{-ix\xi} \, dx. $$
First we take the Fourier transform of the $\operatorname{sign}$ function using $\operatorname{sign}'=2\delta$: $$ 2 = \mathcal{F}\{2\delta\} = \mathcal{F}\{\operatorname{sign}'\} = i\xi \mathcal{F}\{\operatorname{sign}\} . $$ From this we can conclude that $$ \mathcal{F}\{\operatorname{sign}\} = \operatorname{pv}\frac{2}{i\xi} + C\delta(\xi). $$ Here I have used the facts that $u(x)=\operatorname{pv}\frac{1}{x}$ is the odd solution to $xu(x)=1$ and that $x\delta(x)=0.$
But $\operatorname{sign}$ is an odd function, and so must be its Fourier transform, so $C=0.$ Thus, $$ \mathcal{F}\{\operatorname{sign}\} = \operatorname{pv}\frac{2}{i\xi}. $$
Now, $H=\frac12(\operatorname{sign}+1),$ so $$ \mathcal{F}\{H\} = \mathcal{F}\{\frac12(\operatorname{sign}+1)\} = \frac12(\mathcal{F}\{\operatorname{sign}\}+\mathcal{F}\{1\}) = \frac12(\operatorname{pv}\frac{2}{i\xi} + 2\pi\delta(\xi)) = \operatorname{pv}\frac{1}{i\xi} + \pi\delta(\xi) . $$
Then, $$ \mathcal{F}\{xH\} = i\frac{d}{d\xi}\mathcal{F}\{H\} = i\frac{d}{d\xi}(\operatorname{pv}\frac{1}{i\xi} + \pi\delta(\xi)) = -\operatorname{fp}\frac{1}{\xi^2} + i\pi\delta'(\xi) . $$
Finally, $$ \mathcal{F}\{x^2\sin x\} = (i\frac{d}{d\xi})^2 \mathcal{F}\{\sin x\} = -\frac{d^2}{d\xi^2} \mathcal{F}\{ \frac{1}{2i}(e^{ix}-e^{-ix}) \} = -\frac{1}{2i} \frac{d^2}{d\xi^2} 2\pi(\delta(\xi-1)-\delta(\xi+1)) \\ = i\pi (\delta''(\xi-1)-\delta''(\xi+1)) . $$