Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections

Question

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 :

1. 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?
2. Is there a reference for the inequality above?

Answer 1

As pointed by LSpice's comment to the OP, the answer to 1. is yes, for $\{e_k\ |\ k\in\mathbb{Z}^d\}$ is an orthonormal (topological) basis of $L^2(\mathbb{T}^d)$. As for 2., equation (3.33) is one of the so-called Bernstein inequalities, which are probably nowadays more widely known in its continuous version (i.e. for the Fourier transform) and for Fourier series is usually proven for $L^\infty$ norms, see e.g. Theorem 8.2, pp. 49-40 of the book by Y. Katznelson, An Introduction to Harmonic Analysis (third edition, Cambridge University Press, 2002) for the precise version you need (the "reverse Bernstein inequality"), albeit only for the $L^\infty$ norms. The proof in the $L^2$ case (which is the relevant one here) is much easier, thanks to the Parseval fomula. Although it can be inferred e.g. from the Littlewood-Paley dyadic characterization of Sobolev spaces in $\mathbb{T}^d$ (see e.g. Proposition 1.3.1, pp. 11 of R. Danchin's lecture notes) I cannot recall right now a more precise reference for it, though... I will update the answer when I find one.