I've come across a question that looks at things from a different perspective, and I'm not sure how to do it. I have three functions $ f, g, h $ defined in the $ (0, 2 pi) $ Interval, the Fourier series of which are: $$ f (x): sum_ {n = 1} ^ infty {1 over sqrt n} sin (nx) $$ $$ g (x): sum_ {n = 1} ^ infty {1 over n ^ 2 + 1} cos (nx) + {1 over n ^ 4} sin (nx) $$ $$ h (x): sum_ {n = 1} ^ infty {1 over 2 ^ n} cos (nx) $$

Without going into detail because I want to understand the concept first, I am asked a number of questions about belonging to different ones $ L ^ p $Spaces, convergence and so on. The question is … what information can I collect in the Fourier series? What should I pay attention to in order to know my functions?

Thanks a lot!