$\int_{\Bbb{T}} e_n(\lambda) |\varphi(\lambda)|^2 = 0$ for all $n \neq 0$ implies $|\varphi|^2$ is constant almost surely

Question

Consider the circle group $\Bbb{T}\subseteq \Bbb{C}$ with its Haar measure $d \lambda$. I have the following situation: $\varphi \in L^2(\Bbb{T})$ has norm $1$, i.e. $$\Vert \varphi\Vert_2^2 = \int_\Bbb{T} |\varphi|^2 d \lambda=1$$ Put $e_n(\lambda) = \lambda^n, n \in \Bbb{Z}$. We have the following situation $$n \neq 0 \implies \int_\Bbb{T} e_n |\varphi|^2 d \lambda = 0$$

Can I deduce that $|\varphi|^2$ is constant almost surely?

Attempt: I tried to show that $|\varphi|^2 \in L^2(\Bbb{T})$, so that $$|\varphi|^ 2 = \sum_{n \in \Bbb{Z}}\langle |\varphi|^2, e_n\rangle e_n= \langle |\varphi|^2, e_0\rangle e_0$$ by Plancherel's theorem. However, I don't succeed in showing that $|\varphi|^2 \in L^2(\Bbb{T})$

Answer 1

You have $g := |\varphi|^2 \in L^1(\mathbb{T})$ and $$\widehat{g}(n) = \int_{\mathbb{T}} g(x) \cdot \overline{x}^n \, dx = \int_{\mathbb{T}} e_{-n}(x) \cdot |\varphi(x)|^2 \, dx = 0$$ for any $n \in \mathbb{Z}$ with $n \neq 0$. Furthermore $$\widehat{g}(0) = \int_{\mathbb{T}} |\varphi(x)|^2 \, dx = 1.$$ Now check that the constant function $1 \in L^1(\mathbb{T})$ also satisfies $\widehat{1}(n) = 0$ for $n \in \mathbb{Z}$ with $n \neq 0$ and $\widehat{1}(0) = 1$. Then $\widehat{g-1} = 0$ and therefore $g = 1$ almost everywhere by the explanation in your last question.