Differentiability and decay of magnitude of fourier series coefficients

I want to know the answer/references for the question on decay of Fourier series coefficients and the differentiability of a function. Does the magitude of fourier series coefficients {$a_k$} of a differentiable function in $L^2 (\mathbb{R})$ (or in any suitable space) decay as fast as or faster than $k^{-1}$. I want to know if there any such theorem ? Also about the converse statement. ?

There is even a quantitative version of this principle: If $f$ is in $C^r\bigl({\mathbb R}/(2\pi{\mathbb Z})\bigr)$ and if $f^{(r)}$ is of bounded variation $V$ on a full period then the complex Fourier coefficients of $f$ satisfy the estimate $$|c_k|\leq {V\over 2\pi k^{r+1}}\qquad(k\ne0)\ .\qquad(*)$$ In order to prove this for $r=0$ one needs the following

${\it Lemma}.\ $ Let $f$ and $g$ be continuous and $2\pi$-periodic. If $f$ is of bounded variation $V$ and $g$ has a periodic primitive $G$ of absolute value $\leq G^*$ then $$\left|\int_{-\pi}^\pi f(t)g(t)\>dt\right|\leq V\>G^*\ .$$ This is easy to prove by partial integration when $f$ is in $C^1$ and requires some work otherwise. In order to prove (*) for arbitrary $r\geq0$ proceed by induction.

Here are some references:

• Katznelson's Introduction to Harmonic Analysis, p. 22 onwards (He covers the one dimensional case.)
• Grafakos' Classical Fourier Analysis, p. 176 onwards (This covers the torus of arbitrary dimensions case; see below for samplers.)
• This pdf of Braunling's "Fourier Series on the $n$-dimensional Torus".
• That pdf.
• Observe that the correspondence between regularity and decay provided by Fourier transform is also at the core of Sobolev theory, and one might argue Sobolev spaces is the proper setting for this discussion. In this regard the possibilities are vast (e.g. Folland would be a valid starting point).

Here are some samplers from the Grafakos book I list above:

Fix $d\in \mathbb{Z}_{\geq 1}$ as the dimension; and denote the Fourier transform of a function $f:\mathbb{T}^d\to \mathbb{C}$ by

$$\widehat{f}:\mathbb{Z}^d\to \mathbb{C},\quad n\mapsto \int_{\mathbb{T}^d}f(x)e^{-2\pi i\; n\;\bullet\; x}dx.$$

Theorem (Riemann-Lebesgue): $$\forall f\in L^1(\mathbb{T}^d):\lim_{|n|\to\infty} |\widehat{f}(n)| = 0.$$

Theorem (Slow Decay? Sure.): $$\forall d_\bullet: \mathbb{Z^d}\to [0,\infty[: \lim_{|n|\to\infty} d_n = 0,\quad \exists g=g(d_\bullet)\in L^1(\mathbb{T}^d),\forall n\in \mathbb{Z}^d: |\widehat{g}(n)|\geq d_n.$$

Theorem (Decay from Regularity): Let $s\in \mathbb{Z}_{\geq0}$. Then

$$\forall f\in C^s(\mathbb{T}^d):\lim_{|n|\to\infty} (1+|n|^s)|\widehat{f}(n)|=0.$$

Theorem (Regularity from Decay): Let $s\in \mathbb{Z}_{> d}$. Then

$$\forall f\in L^1(\mathbb{T}^d):(1+|n|)^s|\widehat{f}(n)|=O_{|n|\to\infty}(1)\implies f\in C^{s-d}(\mathbb{T}^d).$$

(Here the conclusion is in the sense of "coinciding with a function almost everywhere".)

One convenient soft version of this whole discussion is that a function on the torus is $C^\infty$ iff its Fourier coefficients decay faster than any polynomial.

As a final remark observe the effect of the dimension when going from decay to regularity: Heuristically the growth of the group $\mathbb{Z}^d$ contributes to decay a priori and hence ought to be deducted from the final regularity of the function.