I have been recently working on a periodically driven system and am interested in the window averaged quantities to study the long-time behavior.
Define the window average as
$$\bar{f}(t)=\frac{1}{T}\int_t^{t+T}dt' f(t')$$
Some properties are easy to find, such as $\bar{\dot{f}}=\dot{\bar{f}}$ where $\dot{f}\equiv df/dt$.
I am now particularly interested in $\bar{a\phi}$ where $a$ is a periodic function with period $T$: $a(t)=a(t+T)$ so that I can transform some quick variables to slow (averaged) variables in the ODE I am studying.
$$\bar{a\phi}(t)=\frac{1}{T}\int_t^{t+T}dt'a(t')\phi(t')$$
I have been stuck at this for a while and wonder if there would be a result including simple combination of $\bar{a},\bar{\phi},\bar{\dot{\phi}},\bar{\dot{a}},\text{etc}$?
I'd appreciate any help. Thanks!