Rudin Real and Complex Analysis, Section 4.26

Question

I understood that (6) holds, being a special case of Theorem 4.18. But how do we know that the series on the left converges absolutely?

Answer 1

We have $|\hat{f}| |\overline{\hat{g}}|=|\hat{f}||\hat{g}|\leq \frac{1}{2}(|\hat{f}|^2 + |\hat{g}|^2)$ so \begin{align*} \left|\sum_n \hat{f}(n)\overline{\hat{g}(n)}\right|\leq \sum_n |\hat{f}(n)| |\overline{\hat{g}}(n)|\leq \frac{1}{2}\left(\sum_n|\hat{f}(n)|^2 + \sum_n |\hat{g}(n)|^2\right) \end{align*} The right hand side is equivalent to $\frac{1}{2}(||f||_{L^2(T)}^2 + ||g||_{L^2(T)}^2)<\infty.$