Let $\mathbb{T}$ denote the 1-d torus and $H^s(\mathbb{T})$ the Sobolev space of order $s\geq0$ of complex-valued functions on $\mathbb{T}$ with the identification $H^0 (\mathbb{T}) = L^2 (\mathbb{T})$. Let $\mathcal{T}_N$ denote the span of the first $N \in \mathbb{N}$ trigonometric polynomials and $P_N : L^2(\mathbb{T}) \to \mathcal{T}_N$ the standard Fourier projection. It is easily shown, by passing to sequences, that, for any $f \in H^s (\mathbb{T})$ with $s \geq 1$,

$$\|f-P_Nf\|_{L^2} \leq \bar{N}^{-s}\|f\|_{H^s}$$

where $\bar{N} = \sqrt{1 + N^2}$. Consider now the following simple argument. Let $f \in H^1(\mathbb{T})$ and fix $0 < \epsilon < 1$. Since $C^\infty (\mathbb{T})$ is dense in any $H^s (\mathbb{T})$ space, there exists a function $g_\epsilon \in C^\infty (\mathbb{T})$ such that

$$\|f - g_\epsilon\| < \frac{\epsilon}{2}.$$

Triangle inequality implies

$$\|g_\epsilon \|_{H^1} < \frac{\epsilon}{2} + \|f\|_{H^1}.$$

Since $g_\epsilon \in C^\infty (\mathbb{T})$, we have $g_\epsilon \in H^m (\mathbb{T})$ for any $m \in \mathbb{R}$, and, just to keep this as simple as possible, let's pick $m=2$. Then

$$\|g_\epsilon - P_N g_\epsilon\|_{L^2} \leq \bar{N}^{-2} \|g_\epsilon \|_{H^1} \leq \bar{N}^{-2} ( \frac{\epsilon}{2} + \|f\|_{H^1} ) \leq \frac{\epsilon}{2}$$

simply by picking the smallest $N \in \mathbb{N}$ such that $\bar{N} \geq \left ( \frac{4 \|f\|_{H^1}}{\epsilon} \right )^{1/2}$.

By triangle inequality, we obtain $\|f - P_N g_\epsilon\|_{L^2} \leq \epsilon$.

So, we have shown that, given any $f \in H^1 (\mathbb{T})$ then, for any $N \in \mathbb{N}$, there exists $g_N \in \mathcal{T}_N$ such that

$$\|f - g_N\|_{L^2} \leq 4 \bar{N}^{-2} \|f\|_{H^1}.$$

In fact, we could have done much better and gotten a super-algebraic rate of convergence and, if instead working on $\mathbb{T}^d$, avoided the curse of dimensionality. My immediate question about this (apart from is it correct) is how does one reconcile it with that fact that $P_N$ is the optimal $L^2$-projector onto $\mathcal{T}_N$. In particular, $P_N f$ should be the best one can do when approximating from $\mathcal{T}_N$. This suggests one can do a lot better, provided more regularity is available and is perhaps related to the fact that $H^1$ is relatively compact in $L^2$? Another point of view is that method approximation, particularly $f \mapsto g_\epsilon$ is discontinuous (there are lower bounds that would imply this but they are for non-linear methods and what bothers me most is that $\mathcal{T}_N$ is a linear space). Is the operator of best approximation from $\mathcal{T}_N$ in $L^2$ linear or continuous? Any thoughts on this would be helpful. Thanks.