I am told to prove that $$\sum_{n=2}^\infty \frac{(-1)^n}{n\log(n)}\,e^{inx}$$ is not a Fourier series of an integrable function, and the series $$\sum_{n=2}^\infty \frac{(-1)^n}{n(\log(n))^2}\,e^{inx}$$ is a Fourier series of an integrable function. I have no idea on both of the problems. For the first one, the only two facts that I know to disprove whether a series is a Fourier series are
- The Fourier coefficients must be in $\ell_2$ class; 2. The Abel/Cesaro sum must be bounded.
None of these seems to work in this case. For the second one, those one can use Dirichlet test to show pointwise convergence, I have no idea of proving uniform convergence.