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.

>>mkl+rv
I would count an analytic solution as in the "trying out" category (actually the best kind of trying out!).

The "no other way..." was referring to me not having an intuitive explanation to offer about why an sin(x) and sin(2x) are orthogonal.