Fourier Transformation-exercice

Question

I have this exercice:

Let $$W = \{f \in L^1(\mathbb{R}) : \hat{f} \in L^1(\mathbb{R})\}$$

1- Prove that $f \in W$ iff $\hat{f} \in W.$

2- Let $f \in W.$ $f$ is continuous? $\lim_{x\rightarrow \infty}f(x) = 0$?

Here are my answers. Can you confirm if they are correct please?

1- $f \in W \Rightarrow f \in L^1$ and $\hat{f} \in L^1.$ We have $F(\hat{f}) = \overline{F} (\check{\hat{f}}) = (2 \pi) \check{f}$

wa have $f \in L^1 \Leftrightarrow \check{f} \in L^1,$ so $\hat{\hat{f}} \in L^1(\mathbb{R})$ , so $\hat{f} \in W.$

$f = (2 \pi)^{-1} \overline{F}(\hat{f}) = (2 \pi)^{-1} F (\check{\hat{f}}) \in L^1 = (2 \pi)^{-1} \hat{\hat{f}} \in L^1,$ so $\hat{f} \in L^1$ and $f \in L^1,$ so $f \in W.$

Then the equivalence is prouved.

2- If $f \in L^1$ then $\hat{f}$ is continuous and his limit is 0 in $\infty$.

$f \in W \Leftrightarrow \hat{f} \in W \Rightarrow \hat{f} \in L^1 \Rightarrow \hat{\hat{f}}$ is continuous and $\lim_{x \rightarrow + \infty} \hat{\hat{f}} = 0$ because $f = (2 \pi)^{-1} \check{\hat{\hat{f}}}$

and $\lim f(x) = (2 \pi)^{-1} \lim \breve{\hat{f}} = \lim \hat{\hat{f}} (x) = 0.$

you have $2\pi$? why don't use $\sqrt{2\pi}$ and make it unitary. And what should $\mathcal{F}^2 f$ be?(check wiki.) — newbie, May 23 '13 at 23:25
i do this solution in an exam, i will have zero?? my solution isn't correct? — jijiii, May 24 '13 at 09:54
what is $\bar{F}(\cdot)$? I guess it's the inverse Fourier transform? — newbie, May 24 '13 at 14:14

score 1 · Answer 1 · answered May 26 '13 at 08:50

Yes, the logic of your solution is correct. The details such as multiples of $2\pi$ cannot be completely verified because you did not specify which version of the Fourier transform you are using (there are several commonly used formulas, which differ by the placing of factors $2$ and $\pi$). Regardless of the convention used, the main point is that both direct and inverse Fourier transforms map $L^1$ into $C_0(\mathbb R)$. Therefore, $W\subset L^1(\mathbb R)\cap C_0(\mathbb R)$.

Fourier Transformation-exercice

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