distributional derivative I want to calculate the first and second distributional derivate of the $2\pi$-periodic function $f(t) = \frac{\pi}{4} |t|$, it is
$$
 \langle f', \phi \rangle = - \langle f, \phi' \rangle = -\int_{-\infty}^{\infty} f \phi' \mathrm{d}x 
$$
and
$$
 \langle f'', \phi \rangle = \langle f, \phi'' \rangle = \int_{-\infty}^{\infty} f \phi'' \mathrm{d}x 
$$
so have to evaluate those integrals, but other than $\phi$ is smooth and has compact support i know almost nothing about $\phi$, so it is enough to integrate over finite bound, but how does i use the definition of $f$? So how could i calculate those integrals?

ok, i tried to solve, for the first and second derivate i got
$$
 f'(t) = \frac{\pi}{4}(2H(t)-1)
$$
(where $H(t)$ denotes the Heaviside-Function) on $[-\pi, \pi]$ and $f'(t) = f'(t+2\pi)$. For the second derivate i got
$$
 f''(t) = \frac{\pi}{4}\delta_0(t)
$$
(where $\delta_0$ ist the Dirac-Delta at $0$) on $[-\pi, \pi]$ and $f''(t) = f''(t+2\pi)$. So for the distibutional derivate
$$
 \langle f', \phi \rangle = - \langle f, \phi' \rangle = -\int f \phi' = -\int f'\phi = \langle \frac{\pi}{4}(2H-1), \phi \rangle 
$$
and
$$
 \langle f'', \phi \rangle = \langle \frac{\pi}{4}\delta_0, \phi \rangle
$$
is this right and is my notation acceptable?
