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I felt like it made total sense when I understood the idea of a basis, plus "sinusoids are a basis for (a class of functions)" Everything else basically follo
by ajkjk 2y ago
I felt like it made total sense when I understood the idea of a basis, plus
"sinusoids are a basis for (a class of functions)"
Everything else basically follows from that. The Fourier transform itself integrates (in one notation) e^{-ikx} against f(x). Well, the integral is a giant dot product, e^{-ikx} is the "transpose" of e^{ikx}, one of the basis vectors, so this amounts to saying f_i = <e_i, f> for a basis element e.
- thomasahle 2y agoRight, they are very simple and elegant in algebra. Visualizing them with nested circles flying around and drawing pictures, definitely makes them seem more weird.
- atoav 2y agoThe circles never really bothered me, in the end one circle just gives you the value/time of a partial sine wave. Instead of nesting them you could just put one partial into each row and add them up down below.