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Chapter 2. The Fourier Kingdom - Pg. 33

CHAPTER The Fourier Kingdom 2 The story begins in 1807 when Fourier presents a memoir to the Institut de France, where he claims that any periodic function can be represented as a series of harmonically related sinusoids. This idea had a profound impact in mathematical analysis, physics, and engineering, but it took a century and a half to understand the convergence of Fourier series and complete the theory of Fourier integrals. Fourier was motivated by the study of heat diffusion, which is governed by a linear differential equation. However, the Fourier transform diagonalizes all linear time-invariant operators--the building blocks of signal processing. It therefore is not only the starting point of our exploration but also the basis of all further developments.