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?
distributional derivative
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analysis
functional-analysis
distribution-theory
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2Integrate by parts, and use the fact that $\phi(x)\to 0$ when $x\to \infty$ – 2012-06-23
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0for the first term i got $-\int_{-\infty}^{\infty} f \phi' \mathrm{d}x = -[ f \phi ]_{-\infty}^{\infty} + \int_{-\infty}^{\infty} f' \phi \mathrm{d}x$, the first summand vanishes because of the compact support ($\phi(x) \to 0$ for $x\to \infty$). the derivate of $f$ is $f''(t) = -\pi/4$ for $t \in [-\pi,0]$ and $f''(t) = \pi/4$ for $t\in [0,\pi]$, and $f(t) = f(t + 2\pi)$. so now i have difficulty to bring my knowledge of $f'$ in an derivation of an expression for the derivational derivate? – 2012-06-23
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0Your conclusions is wrong. After you have integrated by patrs you get must get derivative of $f$, not a second derivative. – 2012-06-23
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0oh yes, i am sorry that was a typo, i mean the first derivative $f'$. – 2012-06-23
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0So have you heard of Heaviside function and its derivative? – 2012-06-23
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0yes, Heaviside is the step function, and its derivative is the Dirac delta-function. – 2012-06-23
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0I think the rest is clear. – 2012-06-23