I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an operator $ P_N $ related to Fourier projections on the $n$-dimensional torus $ \mathbb{T}^n $. The operator is defined as: $$P_N u = \sum_{|k|_{\infty} \leq N} \hat{u}_k e_k(x),$$

where $e_k$ are real Fourier basis as stated in Appendix A in the paper. However, the Fourier coefficients, $\hat{u}_k$, are not explicitly stated how they are computed. I would expect it to be the inner product of $u$ and $e_k$.

I am also interested in whether there is a reference in computing the following error which is below Equation (3.33) in the paper: $$ \lVert P_N u - u \rVert_{L^2(\mathbb{T}^n)} \leq \frac{1}{N^s} \lVert u \rVert_{H^s(\mathbb{T}^n)} $$

where $s$ comes from the assumption that $u \in H^s(\mathbb{T}^n)$.

So to summarize :

- Is $\hat{u}_k$ correctly interpreted as the inner product between $u$ and $e_k(x)$ in this Fourier projection context? If not then what is a logical interpretation of it?
- Is there a reference for the inequality above?