What are the limitations /shortcomings of Fourier Transform and Fourier Series?

Question

I am fond of Fourier series & Fourier transform.

But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about

1. The limitations/ shortcomings of the Fourier Series?

2. The limitations/ shortcomings of the Fourier Transform?

Answer 1

Here is my biased and probably incomplete take on the advantages and limitations of both Fourier series and the Fourier transform, as a tool for math and signal processing.

Advantages

• Fourier series and the Fourier transform hold a unique place in the analysis of many linear operators, essentially because the complex exponentials are the eigenvectors/eigenfunctions of linear, shift-invariant operators. In signal processing this is illustrated via the convolution theorem, though the theory goes much deeper (see: Pseduo-differential operators). Related to this is the role of the Fourier transform in the mathematical foundations of quantum mechanics - Fourier analysis is directly related to "momentum", since the eigenfunctions of the momentum operator $-i\partial_x$ are the complex exponentials.
• In this same vein, Fourier analysis leads to an extremely powerful theory of smoothness, because of the correspondence between differentiability and decay of the Fourier coefficients. See Sobolev spaces.
• Fourier analysis is very powerful in the study of generalized functions.
• From a numerical analysis and signal processing point of view, the accuracy of Fourier based methods have the advantage of being limited only by the smoothness of the underlying function. This means several things: Fourier methods are very good at approximating very smooth things, but perhaps not so good at approximating less smooth things. See "disadvantages".
• The general techniques we learn from Fourier, like expanding functions in an orthonormal basis, are extremely powerful. See spectral theory.

Disadvantages

• 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!)

Answer 2

The integral of a canonical Fourier transform must converge, meaning the bandwidth of the signal is somewhat limited. Now consider, the difficulty in interpreting the Fourier transform for even the most common functions, such as cosine, or more interestingly functions like rand(x).