If the series of the Fourier coefficients is absolute convergent then the Fourier series is uniform convergent

Question

It the following correct?

If the series of the Fourier coefficients is absolute convergent then the Fourier series is uniform convergent.

Is it related to some theorem? Parseval's identity?

Answer 1

If the series of Fourier coefficients are absolute convergent, then the series of functions $$\sum_{n=-\infty}^\infty (t\mapsto\hat{f}(n)e^{int})$$ is uniform convergent by Weierstrass M-test, since: $$\sum_{n=-\infty}^\infty \|t\mapsto\hat{f}(n)e^{int}\|_\infty=\sum_{n=-\infty}^\infty |\hat{f}(n)|<+\infty$$