First of all, the question is a failure since it refers to Density Functional Theory or to Discrete Fourier Transform. Which are clearly two completely different topics.

Second, you should define better what defines for you failure. For Density Functional Theory is that comparison with experiments? In the positive, the majority of the papers comparing theory and experiments can be used to explore the failure of the theory. For the Discrete Fourier Transform, this is a more well defined mathematical problem and for sure there are well behaved functions that do not admit a transform.