Let $f$ be a continuous function with diverging partial Fourier sums $S_N(f)(0)$ : $$ f(\theta) = \sum \limits_{k=1}^\infty \alpha_k P_{N_k}(\theta)$$
Let $f(x) \sim \sum \limits_{n=-\infty}^\infty a_n e^{inx} $. Prove that it is possible to change the order of summation to get the series to converge at $x=0$.
How would I go about beginning this proof? Any starting tips would be greatly appreciated because at least I could proceed to work with that then. I am seriously at a loss for words when proof questions take me off track, since then I can't come back to think properly.
Thanks!