# Density and Fourier approximation

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.

I see the glaring error in this now. I applied the bound with the $$H^1$$ norm on $$g_\epsilon$$ when it should have been with the $$H^2$$ norm but then, there’s no way to relate that to the $$H^1$$ norm of $$f$$. Things make sense again. Please close/delete this.