In non-periodic cases, we set fractional Sobolev spaces $ H^s(0,2\pi)$ as

$ H^s(0,2\pi):= \{f \in L^2(0,2\pi): \sum_{n \in \mathbb{Z}} (1+n^2)^s \vert \hat f (n) \vert^2 < \infty \} $

and

$ H^{\infty}(0,2\pi):= \bigcap_{s>0} H^s(0,2\pi) $

define

$\mathcal{U} := \{ u \in H^{\infty} (0,2\pi): \lim_{s\to +\infty} \Vert u \Vert^{\frac{1}{s}}_{H^s} \text{ exists} \}$

and for some positive integer $ p $,

$\mathcal{U}^p := \{ u \in H^{\infty} (0,2\pi): \lim_{s\to +\infty} \Vert u^{(p)} \Vert^{\frac{1}{s+p}}_{H^s} \text{ exists} \} $

Now, the question is

Question 1: What is the accurate characterization of $ \mathcal{U}$ and $ \mathcal{U}^p $ or a subset fulfilled of enough elements?

and further

Question 2: What is the difference between $ \mathcal{U} $ and $ \mathcal{U}^p $?

Notice that $ \mathcal{U} \neq C^{\infty} (0,2\pi) $, use $ u= x \in C^{\infty}$, it does not belong to $ \mathcal{U} $ with a normal examination.

In periodic case, we give notations as

$ H_{per}^s(0,2\pi):= \{f \in H^s(0,2\pi): f(0) = f(2\pi)\} $,

$ H_{per}^{\infty}(0,2\pi):= \bigcap_{s>0} H_{per}^s(0,2\pi) $

and

$ C^{\infty}_{per}[0,2\pi] :=\{ u \in C^{\infty} [0,2\pi]: u(0) = u(2\pi) \} $.

Define

$\mathcal{U}_{per} := \{ u \in H_{per}^{\infty} (0,2\pi): \lim_{s\to +\infty} \Vert u \Vert^{\frac{1}{s}}_{H^s} \text{ exists} \}$

and for some positive integer $ p $,

$\mathcal{U}_{per}^p := \{ u \in H_{per}^{\infty} (0,2\pi): \lim_{s\to +\infty} \Vert u^{(p)} \Vert^{\frac{1}{s+p}}_{H^s} \text{ exists} \} $.

In this case, Question 1,2 could be partially answered that $ \mathcal{U}_{per} = \mathcal{U}_{per}^p$. But

Question3: $C^{\infty}_{per}[0,2\pi] = (\subseteq) \mathcal{U}_{per} $ ?

Notice that $ \emptyset \neq \mathcal{U}_{per} \subseteq \mathcal{U} $. We can examine that $ u:= a_N \cos N t \in \mathcal{U}_{per} $.

Here for question 3, there exists a subtle place. For $ f \in C^{\infty}_{per}[0,2\pi] $, its Fourier series could be termwise differentiated. Then

$ \lim_{n \to \infty} n^k \hat f (n)$ exists for any $ k \in \mathbb{N} $.

Now can it yields that
$ \lim_{k \to \infty} (\sum_{n \in \mathbb{Z}} (1+n^2)^k \hat f (n) )^{\frac{1}{k}} $ exists?

I can vaguely feel the core in this question is the decay rate of Fourier coefficients. Whether can it change from decay of polynomial in any finite order to some more transcendent decay rate?

Any concerned references and thoughts will be welcome!