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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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    See http://math.stackexchange.com/questions/20397/striking-applications-of-integration-by-parts/20545#205452011-03-22

2 Answers 2

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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.

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    could you have a look at http://math.stackexchange.com/questions/30744/one-more-question-about-decay-of-fourier-coefficients, please?2011-04-04
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I think it will be useful for you if you look at the book of Stein & Shakarachi [Fourier Analysis: An Introduction] from p.42 first big paragraph to p.44, and then look at Excercise(10)p.61 in the same book.