Alternative formulation of the Fourier series

Question

Can all real periodic functions be written in the following form?: $$f(x) = a_0 + \sum_{n=1}^\infty a_n \cos\left(\frac{x}{n}\right)+b_n \sin\left(\frac{x}{n}\right)$$

Answer 1

Functions you are describing (provided some convergence) are absolutely not periodical, you should have $ \frac{2 \pi}{\lambda} nx$ as variables for $\lambda$ periodical functions… You can take $a_n = 2^{-n}$ to see for yourself.

Secondly the set of periodical functions is not stable by addition… You need to have rational period to ensure it. And even if you consider that, it is not closed which means infinite sum won’t work.

Lastly sin(2x) is periodical but not in the set you are describing.