Is "Compressed Sensing" compressive? Can it beat the Nyquist Sampling Approach?

Data compression capability of “Compressed sensing (sampling)” in signal discretization is numerically evaluated and found to be far from the theoretical upper bound defined by signal sparsity. It is shown that, for the cases when ordinary sampling with subsequent data compression is prohibitive, there is at least one more efficient, in terms of data compression capability, and more simple and intuitive alternative to Compressed sensing: random sparse sampling and restoration of image bandlimited approximations based on energy compaction capability of transforms. It is also shown that assertions that “Compressed sensing” can beat the Nyquist sampling approach are rooted in misinterpretation of the sampling theory.