# analysis – proof of uniqueness of fourier series in Stein’s book

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.

Posted on Categories Articles