6.1 $U”[x]=\delta'[x+1]+\delta'[x-1]$$

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The Fourier series representation of $U”[x]$ (second-order derivative of $U[x]$) is illustrated below .Note $U”[x]$ corresponds to $\delta'[x+1]+\delta'[x-1]$ where $\delta[x]$ is the Dirac delta function. The first three plots illustrate the evaluation frequencies $f=1$, $f=2$, and $f=3$ respectively and include the left-half plane as well as the right-half plane. The orange reference function in the first three plots below is $\frac{Sign[x]\ f\ \pi}{2\ \log[Abs[x]]}$, which simplifies to $\frac{f\ \pi}{2\ \log[x]}$ in the right-half plane. The fourth plot focuses around $x=1$ and includes multiple evaluation frequencies ($f\in\{2, 4, 8\}$). Note $U”[0]=0$ and $U”[x]$ is an odd function of $x$ (i.e. $U”[-x]=-U”[x]$).


Fourier series representation of $U”[x]$ from $x=-5$ to $x=5$ evaluated at $f=1$.
Fourier series representation of $U”[x]$ from $x=-5$ to $x=5$ evaluated at $f=2$.
Fourier series representation of $U”[x]$ from $x=-5$ to $x=5$ evaluated at $f=3$.


Second-Order Derivative of Unit Step at x=1 Function (i.e. u"[x]) around x=1 Evaluated at f={2, 4, 8}.
Fourier series representation of $U”[x]$ from $x=0.5$ to $x=1.5$ evaluated at $f\in\{2, 4, 8\}$.