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Let $f:\mathbb{R} \to \mathbb{C}$ be a $2\pi -$periodic and continuously differentiable function such that $\hat f(n)=3^{-n^2}$ for $n\in\mathbb{Z},$ where $\hat f(n)$ is the $n-$th Fourier coefficient.

Calculate the Fourier coefficients of $g(x)=\pi f'(x+2009)$

My attempt:

We know the following in general:

$1) \ \hat h'(n)=in \hat h(n)$

$2) \ \hat h(c+x)=e^{inc}\hat h(n), c\in \mathbb{R}$

In our case:

$\hat f'(n)=in\hat f(n)=in3^{-n^2}$

For $t(x)=f'(x+2009): \hat t(n)=e^{in2009}in3^{-n^2}$

So $\hat g(n)=\pi e^{in2009}in3^{-n^2}$

Is my solution correct?

Any help appreciated.

Community
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Itay4
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1 Answers1

1

The proof is absolutely correct. If you want more control on what's going on, you can define some notations:

$$\begin{array}{rcll}{C}_{n} \left(f\right)&=&\widehat{f} \left(n\right)&\text{the fourier coefficient}\\ D \left(f\right)&=&{f'}&\text{the derivation operator}\\ {L}_{c} \left(f\right) \left(x\right)&=&f \left(x+c\right)&\text{the left translation operator} \end{array}$$

Then the rules are

$$\begin{array}{rcl}{C}_{n} \left(D \left(f\right)\right)&=&i n {C}_{n} \left(f\right)\\ {C}_{n} \left({L}_{c} \left(f\right)\right)&=&{e}^{i n c} {C}_{n} \left(f\right) \end{array}$$

Now you have $g = {\pi} {L}_{2009} \left(D \left(f\right)\right)$. The rules give

$$\renewcommand{\arraystretch}{1.5} \begin{array}{rcl}{C}_{n} \left(g\right)&=&{\pi} {C}_{n} \left({L}_{2009} \left(D \left(f\right)\right)\right)\\ &=&{\pi} {e}^{i 2009 n} {C}_{n} \left(D \left(f\right)\right)\\ &=&{\pi} {e}^{i 2009 n} i n {C}_{n} \left(f\right)\\ &=&{\pi} {e}^{i 2009 n} i n {3}^{{-{n}^{2}}} \end{array}$$

Gribouillis
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