The time-domain representation of $G(f) H(f)$, where $H(f)$ is an ideal brickwall filter of bandwidth $1/(2T)$ is:
$$ \int g(\tau) \operatorname{sinc}\left(\frac{t-\tau}{T}\right) d\tau $$
I want to show (using algebra, in the time domain) that the result of the above equation equals $g(t)$ if the bandwidth of $g$ is less than $1/(2T)$. This is something trivial to see in the frequency domain.
But how in the time domain, using convolution integrals?
By definition, if $g$ is bandlimited to $1/(2 T_g)$, I can expand it using the WKS interpolation formula, having the samples as coefficients:
$$ g(t) = \sum_{n=-\infty}^{\infty} g(n T_g) \operatorname{sinc}\left(\frac{t-n T_g}{T_g}\right) = \sum_{n-\infty}^{\infty} g[n] \operatorname{sinc}\left(\frac{t-n T_g}{T_g}\right) $$
Plugging this into the original equation:
$$ \int \sum_{n=-\infty}^{\infty} g(n T_g) \operatorname{sinc}\left(\frac{\tau-n T_g}{T_g}\right) \operatorname{sinc}\left(\frac{t-\tau}{T}\right) d\tau \\ = \sum_{n=-\infty}^{\infty} g(n T_g) \left[ \int \operatorname{sinc}\left(\frac{\tau-n T_g}{T_g}\right) \operatorname{sinc}\left(\frac{t-\tau}{T}\right) d\tau \right] $$
From that, the term in bracket would need to be equivalent to $\operatorname{sinc}\left(\frac{t-n T_g}{T_g}\right)$ if $T \leq T_g$. But I don't see this working out
