The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example:
$g(t,u) \overset{\underset{\mathrm{def}}{}}{=} \int_{t-u/2}^{t+u/2} f(s+\frac{u}{2}) \overline{ f(s - \frac{u}{2}) } ds$
What is the name of this function which does some windowing?
On
The closest thing that comes to mind is a special case of a Wigner distribution function for a signal of finite support.
Looks basically like an autocorrelation (continuous cross-correlation) at lag $u$ to me. See here.