How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?

Question

I have a question about the completeness of complex exponentials in function spaces.

For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, \ldots, n-1 $ forms a basis for $ L^2(S) $, as this is simply a matter of matching dimensions.

However, in the continuous case where $ T = [0, 2\pi] $, the fact that $ e^{ikx} $ for $ k = 0, 1, 2, \ldots $ forms a complete basis (span dense) for $ L^2(T) $ does not seem as intuitive to me. I find the proofs involving the Stone–Weierstrass theorem or Fejér kernels to be quite cumbersome.

Without detailed calculations, I struggle to see why these functions are complete. I would very much like to see a proof using discrete Fourier transform approximation to demonstrate the completeness of $ e^{ik} $ in $L^2(T) $, leveraging the results for $ L^2(S) $. I think such proof would be very intuitive and interesting.

I recently found a related question on Math Stack Exchange (link), where an answer outlines a framework using discrete Fourier approximation to show span density. However, this answer only provides a high-level overview and omits almost all technical details. I am unable to fill in these technical details myself and am even unsure if this framework is reasonable.

The author of this answer has not been active on MSE for almost ten years, so I cannot contact him directly. I would greatly appreciate if someone could help fill in the technical details of this approach or indicate if the method in the answer is not feasible. Additionally, if this approach is based on any existing literature, I would welcome references.

Answer 1

Here is a proof sketch along the lines suggested in the linked SE answer.

I will work on $S^1=[0,1]$ (identifying $0$ and $1$) and denote $e(x)=e^{2\pi ix}$. I will also denote $S_N=\{0,\frac 1N,\ldots,\frac{N-1}N\}\subset S^1$ and assume for convenience that $N$ is odd. Let $f\in C^{2}(S^1)$. Set $f_N(x)=\sum_{n=-(N-1)/2}^{(N-1)/2}\widehat {f_N}(n)e(nx)$, where $\widehat{f_N}(n)=\frac 1N\sum_{x\in S_N}f(x)e(-nx)$. Then $f_N$ coincides with $f$ on $S_N$ by the Discrete Fourier Transform.

A calculation using Abel's summation formula twice and that $f\in C^2$ shows $\widehat{f_N}(n)=O(\frac 1{n^2})$ for $-\frac{N-1}2\le n\le\frac{N-1}2$, uniformly in $N$. This is the discrete version of the (hopefully "intuitive") fact that the Fourier coefficients of a smooth function decay fast. The actual calculation with this approach involves expressions quite similar to Dirichlet and Fejer kernels, evaluated at a discrete set of points. An alternative, more elegant approach is to use the identity $\widehat{\Delta g}(n)=(e(\frac nN)-1)\hat g(n)$ for discrete Fourier transforms, where $\Delta$ is the difference operator $\Delta g(\frac kN)=g(\frac{k+1}N)-g(\frac kN)$.

From this one obtains $\sup_{S^1}|f_N'|=O(\log N)$ and since $f=f_N$ on $S_N$ and the gaps between the points of $S_N$ are $1/N$ we have that $\sup_{S^1}|f-f_N|=O(\log N/N)$ and we are done.