On page 40, the book tries to prove uniqueness of fourier series the following:

Suppose 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 $theta_{0}$.

To prove this, assuming that $f$ satisfies the hypotheses, $theta_{0}=0$, and $f(0)>0$. The idea is now to construct a family of trigonometric polynomials ${p_{k}}$ that peak at $0$, resulting in $$int p_{k}(theta)f(theta); dtheta rightarrow infty ;; text{as};; k rightarrow infty.$$ And this will contradict the requirement that the coefficient needs to be 0.

To construct the p, since $f$ is continuous at $0$, we can choose $0 < delta leq pi/2$, so that $f(theta)>f(0)/2$ whenever $|theta|< delta$. Let $p(theta) = epsilon + costheta$ where $epsilon >0$ is chosen so small that $|p(theta)|< 1-epsilon/2$, whenever $delta leq |theta| leq pi$. Then choose a positive $eta$ with $eta < delta$, so that $p(theta) geq 1 + epsilon/2$, for $|theta| < eta$. Finally, we define $p_{k}(theta) = (p(theta))^k$.

Then the key point will be when $|theta| < eta$,$$int p_{k}(theta)f(theta); dtheta geq 2eta frac{f(0)}{2}(1+frac{epsilon}{2})^k rightarrow infty ; , k rightarrow infty$$

This contracts that the coefficient needs to be 0. So the proof finishes here.

What I’m not understanding is when computing the fourier coefficient using normal formula, $$hat{f}(n)=int f(theta)e^{-intheta}; dtheta.$$ the exponential part should be smaller than 1 $$|p(theta)| = |e^{-intheta}| leq 1$$ so it doesn’t make sense to define a $$|p(theta)| =|epsilon+costheta| gt 1 + frac{epsilon}{2} ; text{when close to 0}$$

How it justifies $p(theta) = costheta + epsilon $. This is the key part since if instead it chooses $p(theta) = costheta – epsilon $ then it won’t contradict the hypothesis as after power of k, it will approaches 0.