It is well known that $\int g(\tau) h(t-\tau)d\tau$ corresponds to $G(\omega) H(\omega)$ in frequency domain.
Can I simplify/say anything (subject to certain assumptions on $g$, $h_1$, $h_2$) about the following integral?
$$ c[n_1,n_2]=\int g(\tau) h_1\left(\frac{n_1 T-\tau}{T}\right) h_2\left(\frac{n_2 T-\tau}{T}\right) d\tau, $$
with $n_1, n_2$ being integers? In particular I would be interested in the following cases:
Case 1
$$ h_1(t)=h_2(t)=\operatorname{sinc}(t). $$
I would "hope" to get zero when $n_1\neq n_2$. Furthermore, I'd hope to get $g(n_1 T)=g(n_2 T)$ if the bandwidth of $g$ is less than $1/(2T)$ (I would get this if there would be only either $h_1$ or $h_2$ by using the convolution theorem of the Fourier transform):
$$ c[n,n] = c[n] = g(nT) . $$
At least this is the result that would intuitively make sense but I don't "see" how I would derive it.
Case 2
$$ h_1(t)=\operatorname{sinc}(t) \\ h_2(t)=\operatorname{rect}(t) $$
For this case, is there any way to simplify this integral? In particular I would like to get rid of either $n_1$ or $n_2$ such that I can write it as $c[n]$ as above.