What are the limitations of fourier theorem
Answers
First off, from a numerical standpoint, issues of convergence play a massive role. See Gibbs Phenomenon. This leads to a secondary issue that Fourier series are not "efficient" at resolving discontinuous or multi-scale functions. This is illustrated, for example, by the vast difference between original JPEG image compression, which is based on Fourier series, and modern image compression techniques like JPEG2000, which are based on more multi-scale techniques like Wavelets.
Related to the above fact is that Fourier series give no information on the spatial/temporal localization of features. A Fourier series or transform can tell you that there is a discontinuity, but it can't tell you where it is. Think of a musical score: having just the Fourier transform is like knowing which notes you need to play, but not when to play them. Not very useful if you want to hear music! This is partially what inspired the study of phase-space/time-frequency/wavelet representations (which incidentally are playing an increasing role in quantum theory).
Classical Fourier analysis is less generally applicable for nonlinear and nonstationary/transient phenomenon (although it is still hugely powerful in some cases!)