Here we discuss taking more samples as a means to more accurately measure the signal; suppose instead we kept the same number of samples, but took more information about each sample (e.g. derivatives at the sample points). What new interpolation strategies does this allow, and what are the drawbacks of this approach?
If I remember correctly from my years as a math undergrad, if we were to take the derivatives of sample points we could use a more complex interpolation approach such as Hermite interpolation. In broad strokes, Hermite interpolation works by using a divided differences algorithm to calculate the coefficients in an interpolation polynomial. For those instances where a divided difference would be undefined, the derivative of the point is used instead. The obvious drawback is that there's a lot more overhead to interpolating this way rather than just using something like a linear interpolation method (or perhaps just taking more samples).