Schwartz Inequality and the Norm Convergence Theorem

Question

I was trying the show that if $f\in PC(2\pi)$ and $x_0 \in [-\pi,\pi]$ then the Fourier series for $f$ can be integrated term by term as $$\int_{x_0}^{x}f(t)dt = \frac{1}{2}a_0(x-x_0) + \sum_{n=1}^{\infty} \int_{x_0}^{x}(a_ncos(nt)b_nsen(nt))dt$$

The problem came with the suggestion that I should consider using the Schwarz Inequality and the Norm Convergence Theorem.

I was able to conclude that $$\left| \int_{x_0}^{x} f(t) dt\right| \le ||f||_2|x-x_0|^2 \le ||f||_2 \sqrt{2\pi}$$ using the suggested Inequality, but I haven't been able to move forward from here.

Thanks in advance.

Answer 1

You forgot the integral on the right side.

Hints: Since $\sum |a_n|^{2} $ and $\sum |b_n|^{2} $ are finite the suggested inequality shows that the series on RHS is uniformly convergent.

Deonte the sum by $g$. LHS and $g$ are both periodic continuous functions. To show that they are equal it is enough to show that they have the same Fourier coefficients. This can shown by an integration by parts.