We consider the function $f$ defined by $$ f(x) = \begin{cases} \dfrac{\sin x}{2} &: x \in ]-\infty,\dfrac{\pi}{2}[\\ 0 &: x \in [\dfrac{\pi}{2},+\infty[ \end{cases} $$ The questions are:
calculate $(T_f)'$,
calculate $(T_f)''$,
deduce the differential equation satisfied by $(T_f)'$ and $(T_f)''$.
My solution is:
$f \in L^1_{loc}(\mathbb{R})$, so we can define the distribution $T_f$ by the relation
$$ \forall \varphi \in \mathcal{D}(\mathbb{R}), \langle T_f,\varphi \rangle = \displaystyle\int_{-\infty}^{+\infty} f(x) \varphi(x) dx= \displaystyle\int_{-\infty}^{\pi/2} \dfrac{\sin x}{2} \varphi (x) dx. $$
- Calclulate $(T_f)'$: we have $$ (T_f)'= T_{f'} + (f(\dfrac{\pi}{2}^+) - f(\dfrac{\pi}{2}^-)) \delta_{\pi/2} = \dfrac{1}{2} \cos(x) + \delta_{\pi/2}. $$
My problem is: How I can calculate $(T_f)''$? with an rule for example, and how we deduce the differential equation satisfied by $(T_f)'$ and $(T_f)''$? Please.
$$ \langle T',\phi\rangle=-\langle T,\phi'\rangle=-\frac12\int_{\pi/2}^\infty(\sin x)\,\phi'(x)\,dx=\frac12\int_{\pi/2}^\infty(\cos x)\,\phi(x)\,dx+\frac12\,\phi(\pi/2). $$ Now for the second derivative: $$\begin{align} \langle T'',\phi\rangle&=-\langle T',\phi'\rangle&\\ &=-\frac12\int_{\pi/2}^\infty(\cos x)\,\phi'(x)\,dx-\frac12\,\phi'(\pi/2)\\ &=-\frac12\int_{\pi/2}^\infty(\sin x)\,\phi(x)\,dx-\frac12\,\phi'(\pi/2). \end{align}$$ Thus $T''=-T+\dfrac12\,\delta_{\pi/2}'$.