Question regarding pointwise convergence of Fourier series

Question

Suppose I have a continuous function $\psi(x)$ on the interval $[0,1]$ and we have $$ \sum_m | \hat \psi (m)| < \infty. $$ Could someone please explain me how it follows that the $\psi(x)$ is the sum of its Fourier series?

Thank you!

Answer 1

The fact that $$ \sum_{m\in\mathbb Z}\big|\hat\psi(m)\big|<\infty, $$ implies that the sequence of the partial Fourier sums $$ \psi_n(x)=\sum_{|m|\le n}\hat\psi(m)\,\mathrm{e}^{imx}, $$ converges uniformly to a continuous function which has Fourier coefficients $\hat\psi(m)$'s and thus coincides with $\psi$ - i.e., $\psi$ is a sum of its Fourier series.