In addition to Robert's more practical answer, I would like to put just few lines on the (indefinetely long and debated) theoretical aspects of auido bandwidth extension.
Since the first days of digital audio compression, people are looking for ways to improve the decoded audio quality. The mass effect came with mp3s and mp4s as you noticed as well. Those codecs throw away the upper parts of the audio spectrum to reduce the compressed bandwidth. Its justification is solidly stated that most people (on the average; a key concept in perceptual lossy codecs) cannot hear the difference. Which is sufficiently verified by tests according to their claims.
So according to this view point, the loss of the audio quality in these codecs do not result from the thrown away spectrum but of the higly nonlinear modifications performed in the remaining audible band. This is especially dependent on the bitrate selected to encode the audio.
Needless to indicate, however, an uncompressed PCM waveform at sufficient bandwidth carries the highest audio quality (say among digital copies) whatever.
Therefore, algorithms still exist claiming to recover the missing spectrum. But is that really possible? All of these algorithms are clever. They add new spectrum that will feel pleasant. Just like old analog equipment adding pleasant distortions.
Furthermore, some more serious of them try to extend the bandwidth consistently with the existing spectrum. This demands a more mathematical treatment though, harmonic extensions, dependencies, and even physical constraints are considered; instrument modeling could be an ultimate tool in extending the harmony for example.
But eventually, one must recognize that, especially at mediocre compression levels the existing auido is highly modified and most of the subtle virtues are lost, and only regenerated to be similar in feeling to the original for the average listener. All of those algorithms, as a consequence, rely on this available compressed approximation to reconstruct the missing band, which may create an even a harder problem than the original at some instances.
Can you even reconstruct the audible band perfectly ?