| For his analytic theory of heat, he was working with differential equations to express heat propagation. Those equations are especially easy when applied to trigonometric functions like sine and cosine (because their derivates are easy to express in terms of each other). He didn't really have a way to solve the equations for other functions directly, but he tried to approximate them by sums of trigonometric functions. He noticed that he could actually do that for all functions he wanted to analyze, and wrote about that in his book. Later, other mathematicians qualified the circumstances that make such a transformation possible, formally proved its properties and named it "Fourier transform" in his honor. Because of its applicability to problems involving differential equations, the Fourier transform was also used in other domains of physics e.g. to analyze vibrations like sound. Because of those many uses, people needed a way to compute it quickly, so they optimized the algorithm to create the Fast Fourier Transform. So those "tools that are so important in digital signal processing today" were not invented by Fourier singlehandedly, but are associated with his name due to him kicking off the initial development. Actually, Gauss used a variant of the FFT in his astronomical work even before Fourier published his own results, but because Gauss didn't publish, he is not associated with the discovery. |