Distributional order in terms of Fourier coefficients growth rate

Question

Let $f$ be a generalized function over the $d$-torus $T^d:=(\mathbb R/2\pi\mathbb Z)^d$, defined by its distribution $L_f\in C^\infty(T^d)^*$. Since $T^d$ is compact, $f$ has well-defined Fourier coefficients $$ \hat f:\mathbb Z^d\to\mathbb C: \mathbf n\mapsto L_f(\psi_{-\mathbf n}) $$ where $\psi_{\mathbf n}:T^d\to\mathbb C:\mathbf q\mapsto e^{i\mathbf n\cdot\mathbf q}$.

I was wondering whether the behaviour of the Fourier coefficients as $\Vert\mathbf n\Vert\to\infty$ could tell us what kind of generalized function $f$ is. In particular, I am looking for characterizations of the following type:

1. $|\hat{f}(\mathbf{n})| \sim \mathcal O(\Vert\mathbf{n}\Vert^{-(k+2)}) \quad$ if $ f\in C^k(T^d) $
2. $|\hat{f}(\mathbf{n})| \sim \mathcal O(\Vert\mathbf{n}\Vert^{-1}) \hspace{24pt}$ if $ f\in L^2(T^d)$
3. $|\hat{f}(\mathbf{n})| \sim \mathcal O(\Vert\mathbf{n}\Vert^{0}) \hspace{30pt}$ if $ df:$ finite Borel measure on $T^d$
4. $|\hat{f}(\mathbf{n})| \sim \mathcal O(\Vert\mathbf{n}\Vert^{k}) \hspace{29pt}$ if $ L_f\in C^k(T^d)^*$

Are such results known? If so, how would one go about proving these?

My first guess would be to look at the Riesz–Fischer theorem for point 2, and the Riesz–Markov–Kakutani representation theorem for 3.

Answer 1

This is an interesting question. I do not have a clear-cut answer, hence my post is community wiki.

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Concerning the $C^k$ class, see this question. However, there the subject is the Fourier transform on $\mathbb R$.

See also "An introduction to harmonic analysis" of Katznelson, section 1.4, "The order of magnitude of Fourier coefficients".

Concerning the $L^p$ class, I think that the only neat answer is the theorem of Riesz and Fisher: $$f\in L^2 \quad \iff \quad\hat{f}\in \ell^2.$$ I am sure that the belonging to any other $L^p$ class other than $L^2$ cannot be determined by the moduli of the Fourier coefficients alone; this is due to the example of Zygmund $$ f_\omega(x)=\sum_{n=-\infty}^\infty \epsilon_n \hat{f}(n)e^{i n x}, $$ where $\hat{f}(n)\in \ell^2$ and $\epsilon_n$ is a sequence of iid Bernoulli random variables (that is, $\epsilon_n=+1$ or $\epsilon_n=-1$ with equal probability $1/2$). Since $$\mathbb E\left( \int_{\mathbb T} |f^\omega(x)|^{2k}\, dx \right)<\infty\quad \forall k\in\mathbb N, $$ this function is, almost surely, in $L^p(\mathbb T)$ for all $p\in[1, \infty)$; this is nontrivial, but elementary. (I like how this is explained in these lecture notes of Nicolas Burq).

So, just shuffling at random the signs of the Fourier coefficients of a $L^2(\mathbb T)$ function, almost surely we get an $L^p(\mathbb T)$ function. All without touching the moduli of these Fourier coefficients. The conclusion is: there is no hope for a criterion that identifies the $L^p(\mathbb T)$ class of a function in terms of the moduli of the Fourier coefficients alone.