I am currently reading this paper which makes use of generalized differential operators.
As I understood it, the operator $D_x$ works like this: If $F$ is a continuous function on $[a,b]$ and $G$ an (integrable?) function on $[a,b]$ then $D_x F = G$ means that there is a constant $c \in \mathbb{C}$ such that
$F(x) = c + \int_a^x G(z) dz$.
(So actually quite ordinary, or did I miss something?)
My problem now is the following: At some point in this paper the author assumes $F$ to be a piecewise linear function on some interval $[0,l]$. She then takes $x_1, x_2 \in [0,l]$ and supposes that $F$ is linear between $x_1$ and $x_2$ and states the following:
$\int_{x_1}^{x_2} D_s F(s)^2 ds = \frac{\left[ F(x_2)-F(x_1) \right]^2}{x_2 - x_1}$
And I do not get that at all.
I mean, if my assumption was correct and $D_s$ actually means differentiation w.r.t. $s$ then $\int_{x_1}^{x_2} D_s F(s)^2 ds$ should be $F(x_2)^2-F(x_1)^2$, right?
Maybe someone knows the proper definition of this $D_s$ and can help me, please?