zlacker

>>qqqqqu+(OP)
Fourier Transforms. I'd wish I had a intuitive understanding of how they work. Until then I'm stuck with just believing that the magic works out.

>>rambol+mj
The best way to understand the Fourier transformations is to think of them as change-of-basis operations, like we do in linear algebra. Specifically a change from the "time basis" (normal functions) to the "frequency basis" (consisting of a family of orthonormal functions).

Here is the chapter on Fourier transforms from my linear algebra book that goes into more details: https://minireference.com/static/excerpts/fourier_transforma...

As for the math, there really is no other way to convince yourself that sin(x) and sin(2x) are orthogonal with respect to the product int(f,g,[0,2pi]) other than to try it out https://live.sympy.org/?evaluate=integrate(%20sin(x)*sin(2*x... Try also with sin(3x) etc. and cos(n*x) etc.

>>ivan_a+6o
> As for the math, there really is no other way to convince yourself that sin(x) and sin(2x) are orthogonal with respect to the product int(f,g,[0,2pi]) other than to try it out https://live.sympy.org/?evaluate=integrate(%20sin(x)*sin(2*x.... Try also with sin(3x) etc. and cos(n*x) etc.

I disagree with that. It's pretty easy to prove it in general by calculating \int_0^{2\pi} sin(mx)sin(nx) dx etc. for m ≠ n.