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. ?

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possible duplicate of Decay of Fourier coefficients – t.b. Mar 22 '11 at 9:12
@Theo Buehler : I thought this is primary step to answer the other questions and i am not able to. – Rajesh D Mar 22 '11 at 11:39
I see. I don't have time now but I might answer the other question in the next few days. I'll notify you if that should happen. – t.b. Mar 22 '11 at 11:43
But for the moment I can give you a hint: Use Fejér's theorem: If $f$ is a continuous function of bounded variation then the Fourier series converges uniformly to $f$. Then think about what differentiation does to the Fourier series. – t.b. Mar 22 '11 at 12:03
– David Speyer Mar 22 '11 at 13:13

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.