A periodic function $f(x)$ with period $2\pi$ can be written as: $$f(x) = \frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos{nx}+b_n\sin{nx}$$ where, $\cos{nx}$ and $\sin{nx}$ are two basis functions. The weights for the basis, $a_n$ and $b_n$ are defined as: $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi}f(x)\cos{nx}\,dx\\b_n = \frac{1}{\pi} \int_{-\pi}^{\pi}f(x)\sin{nx}\,dx$$ (a) Prove/Disprove that the bases are linearly independent.
(b) Prove/Disprove that the bases are orthogonal.
(c) Compute the weights for the bases for the function $f(x) = \cos{2x}$.
I just know the basics what being orthogonal and being linearly independent means, I have not solved any such problems before, nor I have any reference book to study/learn/practice material like this one.
Please provide solution to this, as well as reference books or online courses or any relevant content using which one could learn to solve such questions.
a friend of mine, proved two parts of this question, left is the part (c)
