I am brushing up on Functional Analysis using the video lectures and problem sets by Casey Rodriguez (see here) and came across a problem which does not look extremely hard, but which I cannot quite solve. It gives a characterization of the Sobolev Space $H^s(\mathbb{T})$, and asks to prove that some condition on a function $f$ is sufficient for $f$ to be in $H^s(\mathbb{T})$. Some research taught me that Sobolev Spaces are a big topic with lots of theory to study and useful applications as well, but any knowledge about Sobolev Spaces in general should not be required for the exercise. The problem is as follows:
Let $s \geq 0$. We say that $f \in L^2([-\pi, \pi])$ is an element of the Sobolev Space $H^s(\mathbb{T})$ of order $s$ if $$ \lim_{N\to\infty} \sum_{|n|\leq N} |\hat{f}(n)|^2(1+|n|^2)^s < \infty. $$ Suppose that $f \in C^k([-\pi, \pi])$ and $f(\pi) = f(-\pi), f'(\pi) = f'(-\pi), ..., f^{(k-1)}(\pi) = f^{(k-1)}(-\pi)$. Prove that $f \in H^s(\mathbb{T})$ for all $0 \leq s \leq k$. Hint: Integrate by ...
Wherein the Fourier coefficients $\hat{f}(n)$ are defined as $\hat{f}(n) = \frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int}dt$ (using Riemann notation as in this case $f$ is continuous, so the Riemann integral agrees with Lebesgue)
Now, for my attempt at a solution, I presumed that the hint alluded to integration by parts. For the fourier coefficients $\hat{f}(n)$ of $f$ we find, by repeated application of integration by parts: $$ \hat{f}(n) = (-\frac{i}{n})^k \int_{-\pi}^\pi f^{(k)}(t)e^{-int} dt $$ so we see that for all $n \in \mathbb{Z}$ we have $$ |\hat{f}(n)| \leq \frac{1}{|n|^k}||f^{(k)}|| $$ which is a real number since $f^{(k)}$ is continuous and therefore bounded on $[-\pi, \pi]$. Now, we want to use this to check if $f \in H^s(\mathbb{T})$:
(I pasted a screenshot from another latex editor here because stackexchange didn't compile properly) After which I wanted to prove that the infinite series converges. this might be true for $s$ strictly smaller than $k$, though I wasn't able to prove it, but it's clearly false if $s=k$, which leads me to believe that choosing this upper bound for the fourier coefficients is not the way to go. Did I make a mistake somewhere, or is there a better approach I'm missing? Should I find a bound for all $N \in \mathbb{N}$ instead of trying to bound the infinite series as $N \to \infty$? Any help would be greatly appreciated.
