Joseph Fourier announced in 1807 that any 27π-periodic function could be represented as a series of sines and cosines. Since then, the spectral analysis of functions using Fourier series and integrals has been the source of numerous mathematical problems. Examples include the efforts to characterize function spaces in terms of their spectral properties and the study of operators on these spaces. In general, the problems encountered in these programs are related to the fact that it is impossible to describe the local properties of functions in terms of their spectral properties, which can be viewed as an expression of the Heisenberg uncertainty principle. Other problems arise because Fourier series may diverge in many of the usual function spaces.
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