# Estimate for the Fourier coefficients

In the book I am reading, the author gives an estimate for the Fourier coefficients and I don't get it.

In the following we consider integrable functions $$f \in L^1(S^1)$$ which are measurable functions $$x \mapsto f(e^{2􏰃 \pi ix})$$, periodic of period 1 and, abbreviating $$f(e^{2 \pi 􏰃ix}) =􏰌 f(x)$$, satisfy

$$\int_0^1 \mid f(x) \mid dx < \infty.$$

The Fourier coefficients are defined as

$$\widehat{f}(n) = \int_0^1 f(x)e^{-2 \pi inx} dx.$$

The statement is that for every integer $$k \geq 0$$, there exists a constant $$c_k$$ such that

$$\mid \widehat{f}(n) \mid \leq \frac{c_k}{\mid n \mid^k}.$$

Since there's this $$\mid n \mid^k$$ in the denominator, I tried to use integration by parts repeatedly, but unfortunately I don't know how to conclude. Here's my attempt:

I derivate $$f$$ and integrate $$e^{-2 \pi inx}$$. The boundary terms cancels out, because of the periodicity, hence I get

$$\widehat{f}(n) = - \frac{i}{2 \pi n} \int_0^1 f'(x)e^{-2 \pi inx} dx$$

Is it the right path to follow ? After 3 integration by parts, I get

$$\widehat{f}(n) = \frac{-i}{(2 \pi n)^3} \int_0^1 f'''(x)e^{-2 \pi inx} dx.$$

So I would get the following estimate

$$\mid \widehat{f}(n) \mid \leq \frac{1}{(2 \pi n)^3} \int_0^1 \mid f'''(x) \mid dx.$$

Which seems to be near to what is stated but not quite. How can I conclude ?

Many thanks for your kind help.

• You left out a crucial hypothesis - that estimate is not true for every $f\in L^1(S^1)$. – David C. Ullrich Mar 9 at 18:36
• Take $C_k = \frac{1}{(2\pi)^k}\int_{0}^{1}f^k$. Also you need some sort of smoothness assumption. – rubikscube09 Mar 9 at 18:36
• @DavidC.Ullrich Thanks you. You're right, the author take $f \in C^{\infty}(S^1,\mathbb{R}^n)$. – Alain Mar 9 at 18:40
• @rubikscube09 but how can I be sure that $\int_0^1 f^{k}$ is fine ? Do I miss something ? – Alain Mar 9 at 18:41
• It seems ok to me. It is a constant that depends only on $k,f$, and is well defined thanks to the smoothness of $f$. – rubikscube09 Mar 9 at 18:49