Respuesta :
Apparently, some people claim that every signal may be described as a Fourier series, notably in electrical engineering and musical signal processing.
This prompted me to consider the mathematical justification for such an argument.
However, even after reading through various materials on the Fourier series (which I have little background in but understand the notion of), I was unable to locate a mathematical justification for the claim that every function can be represented by a Fourier series. The need for the function to be periodic was alluded to.
Most functions cannot be expressed as Fourier series, as may be seen by a simple counting argument. A countable family of Fourier coefficients serves as the basis for a Fourier series, and as a result, the set of such series has cardinality c0, as opposed to the set of real valued functions defined on some interval, which has cardinality cc.
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