A naive (and in my opinion wrong) approach to doing the interpolation is to linearly interpolate between the samples. This does work reasonably well for low frequency sounds but doesn't work at all well for high frequency sounds.

Here we have a sine wave (read) sampled at the location of the blue dots. The green dots are new values which are linearly interpolated between the samples represented by the blue dots.

If we then attempt to reconstruct the the waveform from the original samples (blue) and the interpolated samples (green) we don't get back a sine wave.

On more complex waveforms the problem is far worse. Interpolation using polynomials is not the solution either.