Comment by behnamoh
9 hours ago
This gets brought up quite often here but something people don't talk about is why Fourier needed to do this. Historical context is really fun! In the late 1800s, Fourier wanted to mathematically describe how heat diffuses through solids, aiming to predict how temperature changes over time, such as in a heated metal rod. Observing that temperature variations evolve smoothly, he drew inspiration from the vibrating string problem studied by Euler and D’Alembert, where any complex motion could be expressed as a sum of simple sine waves. Fourier hypothesized that heat distribution might follow a similar principle; that any initial temperature pattern could be decomposed into basic sinusoidal modes, each evolving independently as heat diffused.
Minor correction, Fourier made his breakthroughs in the early 1800's. He worked under the reign of Napoleon and continued in the decade thereafter.
And it is an important correction because Fourier was immediately able to use his technique to solve partial differential equations, but it was many decades later before it was shown how it all works with the rigorous foundation of measure theory and functional analysis.