(in the second line, the exchange of limiting operations is justified by the uniform convergence of the series. In the next line, I made the substitution and used the fact that .)
I now want to complete the proof by saying that g is the sum of its Fourier series: . To justify that, I need to quote some theorem saying that g is the sum of its Fourier series. The only such result that I know says that this holds if g is continuous. But the given information does not seem to ensure that g is continuous. Perhaps we need to be told that F is continuous? That would imply that g is continuous, because of the uniform convergence.