what are the limitations of fourier theorem?
Answers
Fourier series have the benefit of being discrete which makes it easy to do computationally. However it requires that your signal be on a finite domain. In practice this isn't a problem so much. However the functional analytic properties of Fourier series are not that nice. Fourier transforms deal with signals that don't have compact support and can be thought of as a translation between functions of the same type: it's a unitary map on an inner product space. Fourier series don't have this property which makes them so much harder to study in full detail.
Methods based on the Fourier transform are almost synonymous with frequency domain processing of signals (funnily, I once had a classmate who thought “Fourier” was French for frequency). There is no doubt about how incredibly powerful Fourier analysis can be. However, its popularity and effectiveness have a downside. It has led to a very specific and limited view of frequency in the context of signal processing. Simply put, frequencies, in the context of Fourier methods, are just a collection of the individual frequencies of periodic signals that a given signal is composed of. The purpose of these tutorials is to demonstrate how restrictive this interpretation of frequency can be in some cases, and to lay the groundwork for complementary methods, like the Hilbert spectral analysis.
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