Reference request for the relation between smoothness and magnitude of Fourier coefficients

Question

The following statement can be found for example in these lecture notes (Theorem 1.7 and 1.9) with proofs but without a concrete reference:

Let $n \in \mathbb{N}_0 $, denote the Fourier coefficients of a function $f \in L^2(-\pi,\pi)$ by $f_k$, let $C^n$ be the set of $n$ times continuously differenciable functions. Then:

(i) $|f_k| < \frac{C}{|k|^{1+n+\epsilon}}$ for some $C,\epsilon>0$ $\Longrightarrow$ $f \in C^n$

(ii) $f \in C^n$ $\Longrightarrow$ $|f_k| \leq \frac{C'}{|k|^n}$ with $C' = \sup_x |\tfrac{d^n}{dx^n}f(x)|$

These should be pretty standard results. I have tried to find a good textbook on Fourier analysis that contains these statements, but haven't found any good reference so far. Any recommendations?

Answer 1

I suggest An Introduction to Harmonic Analysis by Katznelson. In the 2nd edition, statement (ii) is Theorem 4.4, and statement (i) appears as Exercise 2 of the same section. There is a 3rd edition now, which is substantially expanded.

Answer 2

There is an exact characterization of differentiability for the $L^2$ case. This happens because $e^{inx}$ is an orthonormal basis of the selfadjoint operator $L=\frac{1}{i}\frac{d}{dt}$ defined on the natural domain $\mathcal{D}(L)$ consisting of all periodic absolutely continuous functions $f\in L^2$ for which $f'\in L^2$. You can prove directly that $f \in L^2$ is absolutely continuous with $f'\in L^2$ iff $\sum_{n=-\infty}^{\infty}n\hat{f}(n)e^{inx} \in L^2$, which holds iff $\sum_{n=-\infty}^{\infty}n^2|\hat{f}(n)|^2 < \infty$.

Therefore, $$ f\in\mathcal{D}(L^s) \iff \sum_{n=-\infty}^{\infty}n^{2s}|\hat{f}(n)|^2 < \infty,\;\;\; s=1,2,3,\cdots. $$ And $f\in \mathcal{D}(L^s)$ iff $f$ has $s-1$ continuous derivatives and $f^{(s-1)}$ is absolutely continuous with $f^{(s)}\in L^2$.

Once you get away from $L^2$, there is nothing so nice. But this is an if and only if result, which may make it useful to you.