What do you mean? If the original source is analog and sampled below its Nyquist rate in the analog-to-digital conversion, the process is indeed irreversible. But that all happens before any transforms from the time domain to the frequency domain are in play, so it's a separate issue.

Beyond that, discrete Fourier and cosine transforms as usually implemented are not fully reversible because of loss in precision. A colleague of mine blogged about the issue in the context of Haar transforms a few years ago: http://cbloomrants.blogspot.com/2008/09/09-08-08-1.html. By decomposing an orthogonal transform into shears as explained by Charles, you can design reversible fixed-precision variants of the DCT like binDCT: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.8...

Sorry it is a separate issue - I should have been more clear.

The original point the OP made was a bad example. Unfortunately I made a poor attempt at explaining that.