I was doing some exercises but I have some doubts. Here are some of the problems that I'm facing:
- Compute the F.T. of $\mathcal{F}D^kT$, where $D^k$ is the $k$-th derivative of the distribution $T$.
I did the following steps: let $\phi(x)\in S(\mathbb{R})$, and $\hat\phi(y)$ its F.T. : $$ \begin{align} (\mathcal{F}D^kT)[\phi] &=(D^kT)_{\mathcal{F}}[\phi]\\ &=(-1)^kT_{\mathcal{F}}\left[\left(\frac{d}{dx}\right)^k\phi\right]\\ &=(-1)^kT\left[\mathcal{F}\left(\frac{d}{dx}\right)^k\phi\right] \end{align} $$ and by the property $\mathcal{F}[(d/dx) f] = -iy\mathcal{F}[f]$: $$ \begin{align} (-1)^kT\left[\mathcal{F}\left(\frac{d}{dx}\right)^k\phi\right] &= (-1)^kT\left[(-iy)^k\mathcal{F}[\phi]\right]\\ &= (iy)^kT[\mathcal{F}[\phi]]=(iy)^k\mathcal{F}[T][\phi] \end{align} $$ But the right answer is: $(-iy)^k\mathcal{F}[T]$ (with the minus sign in front of the $i$). Where is the mistake?
- Compute the F.T. of $D^k\mathcal{F}[T]$, where $D^k$ is the $k$-th derivative. As before $\phi(x)$ is a function of $x$ and $\hat\phi(y)$ is a funcion of $y$. My calculations are: $$ \begin{align} (D^k\mathcal{F}[T])[\phi]&=D^k(\mathcal{F}[T])[\phi]\\ &=(-1)^k(\mathcal{F}[T])\left[\left(\frac{d}{dx}\right)^k\phi\right]\\ &=(-1)^kT\left[\mathcal{F}\left(\frac{d}{dx}\right)^k\phi\right] \end{align} $$ Like before i used the properties for the derivative inside the F.T., $$ \begin{align} (-1)^kT\left[\mathcal{F}\left(\frac{d}{dx}\right)^k\phi\right]&=(-1)^kT\left[(-iy)^k\mathcal{F}[\phi]\right]\\ &=(-1)^kT\left[\mathcal{F}\left(\frac{d}{dx}\right)^k\phi\right]\\ &=(iy)^kT\left[\mathcal{F}[\phi]\right]=\mathcal{F}[(iy)^kT][\phi] \end{align} $$ I'm not so sure about the last equality. I used $T_1 = ((iy)^kT)$ and $T_1[\mathcal{F}[\phi]] = \mathcal{F}[T_1][\phi]$ is that even a thing?
Also there are other things that confuse me:
- When i have $(FD^kT)$ and I want to apply it to a function, do I write this $(FD^kT)[\phi]$ or this $FD^k(T[\phi])$? What is the correct order of computation?
- Is it correct to say $(FD^kT)=F(D^kT)=(-iy)^kF(T)$?
Thanks in advance for any response, also pls correct any grammar mistakes, I'm not mother tongue :)
