Energetic Identities, Manipulations and Inequalities

Of extreme utility in energy analysis are various manipulations which correspond to integration by parts. For example, consider the inner product $\langle f,\delta_{x+}g\rangle_{[r,s]}$, where, for brevity, the time index $n$ has been dropped from the grid functions.

$$\langle f,\delta_{x+}g\rangle_{[r,s]} = \sum_{l=r}^{s}hf_{l}\frac{1}{h}\left(g_{l+1}-g_{l}\right)$$ (5.13)