My notes when reading Fourier Analysis An Introduction by Stein and Shakarchi. This is for chapter 2 Basic Properties of Fourier Series, section 2 Uniqueness of Fourier series.

2 Uniqueness of Fourier series

Thereom 2.1 Supporse that $$f$$ is an integrable function on the circle with $$\hat{f}(n)=0$$ for all $$n\in\mathbb{Z}$$. Then $$f(\theta_0)=0$$ whenever $$f$$ is continuous at the point $$\theta_0$$.

Corollary 2.2 If $$f$$ is continuous on the circle and $$\hat{f}(n)=0$$ for all $$n\in\mathbb{Z}$$, then $$f=0$$.

Corollary 2.3 Suppose that $$f$$ is a continuous function on the circle and that the Fourier series of $$f$$ is absolutely convergent, $$\sum_{n=-\infty}^\infty \lvert\hat{f}(n)\rvert <\infty$$. Then, the Fourier series converges uniformly to $$f$$, that is

$$ \lim_{N\to\infty} S_N(f)(\theta)=f(\theta)\quad\mathit{uniformly\:in}\:\theta\mathit{.} $$

Corollary 2.4 Suppose that $$f$$ is a twice continuously differentiable function on the circle. Then

$$ \hat{f}(n)=O(1/\lvert n\rvert ^2)\quad as \:\lvert n\rvert\to\infty $$

so that the Fourier series of $$f$$ converges absolutely and uniformly to $$f$$.¹

Hölder Condition: the Fourier series of $$f$$ converges absolutely (and hence uniformly to $$f$$) if $$f$$ satiesfies a Hölder Condition of order $$\alpha$$, with $$\alpha >1/2$$, that is,

$$ \sup_\theta\lvert f(\theta +t)-f(\theta)\rvert\leq A\lvert t\rvert^\alpha\quad\mathrm{for\:all}\:t\mathrm{.} $$

Footnotes