CHAPTER 2 The Fourier Kingdom

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 ...

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