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

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?
• Yes for (1). And some more characters. Commented May 7 at 21:54

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.
• Sorry for the delay in getting back to you. The Bernstein inequality in the OP essentially follows in the case of $\mathbb{R}^d$ from the second inequality in Proposition 1.3.1 in the case $N=2^n$ if we remove the first $n$ terms from the sum in the lhs, up to normalization terms. In the case of $\mathbb{T}^d$, the argument is exactly the same but one then replaces the Littlewood-Paley projections $\Delta_q u$ with its Fourier components $\langle e_k,u\rangle e_k=\hat{u}_k e_k$. Prop. 1.3.1 in that case then follows from integration by parts and the Parseval formula. Commented May 29 at 20:09