Bandwidth expansion by computing missing harmonics in music?
Bandwidth expansion is actually a problem of missing audio data recovery, which is an active area of research.

Usually you assume that your signal follows some statistical model in the time-frequency domain, and the whole challenge is to infer the parameters of your model, based only on the available data.

In the case of bandwidth expansion, you would infer your model from the lower frequency part of your spectrogram, in order to recover the higher frequencies.

You can have a look at some of Paris Smaragdis' work on missing audio data and bandwidth expansion, using latent variable models. Some demos are available here.