Suppose that a continuous signal $f(t)$ has the first harmonic frequency $f_1$. $f(t)$'s frequencies that are not integer multiples of $f_1$ are known to have zero signal magnitude $|F(\omega)|$. This does not mean that every integer multiples of $f_1$ have non-zero signal magnitude - some of them may have zero magnitude.
Now the highest possible frequency $f_h$ of $f(t)$ is also known - so the integer multiples of $f_1$ greater than $f_h$ all have zero signal magnitude. But this also does not mean that every integer multiples of $f_1$ below $f_h$ have non-zero signal magnitude. Some of them may have zero signal magnitude.
It is also known that all frequency magnitude be non-negative.
With these knowledge, would there be a way to sample $f(t)$ with finite number of samples $x_1,...,x_m$ with $m<n$ where $f_h = nf_1$ to transform $f(t)$ into $F(\omega)$ with good degrees of approximation? Because in DFT you only get $m$ non-zero frequencies if the number of samples is $m$. (And possibly invert back into $f(t)$?)