This is Exercise 1 of Chapter 8 in Rudin's Functional Analysis. We are asked to prove the following:

If $P$ is a polynomial in $\mathbb{C}^n$ and if \begin{equation}\int_{T^n}|P|d\sigma_n=0,\end{equation} then $P$ is identically zero.

Here $T_n$ is the n-dimensional torus and $\sigma_n$ is the Haar measure on $T_n$.

It should not be difficult but I guess I am missing a trick in complex analysis.



If you restrict $P$ as a function $\mathbb{T}^n$, then the Fourier series for $P$ is simply the polynomial itself (up to constant maybe). Therefore, by Plancherel's theorem, we have $$ \int_{\mathbb{T}^n} |P| = 0 \Longrightarrow \int_{\mathbb{T}^n} |P|^2 = \sum_{j} |a_j|^2 = 0$$ again with possibly a constant $C > 0$ in front, and the $a_j$'s are the Fourier coefficients of $P$, which are the coefficients of the polynomial! Therefore $a_j = 0$ for all $j$, hence $P \equiv 0$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.