Can we ever reconstruct a a signal by its sampling? No, we generally never can but we can approximate.
1. The easiest way to "reconstruct" a signal is by zero-order interpolation which looks like step functions.
$ x(t) = \sum^{\infty}_{k = -\infty} x(kT) (u[t-kT]-u[t-(k+1)T]) $
2. To step it up we can use 1st order interpolation. She gave an example about a kid going to an interview and they asked him if he has ever heard of splines and peace-wise polynomial functions and that is what this is.
$ x(t)= \sum^{\infty}_{k = -\infty} x(t_k) + (t-t_k \frac {x(t_{k+1})-x(t_k)}{t_{k+1} - t_k}) $
and $ \frac {x(t_{k+1})-x(t_k)}{t_{k+1} - t_k} $ is just the slope.
And as you can see, the smaller the sampling is, the better chance you have of what the signal looks like.
