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Assume that $ f\in L^1(\mathbb{R}) $ and $ f $ is continuous at the point $ 0 $. If $ \widehat{f}(\xi)=\int_{\mathbb{R}}f(x)e^{-2\pi ix\xi}dx\geq 0 $ for any $ \xi\in\mathbb{R} $, show that $ \widehat{f}\in L^1(\mathbb{R}) $.

It can be get that $$ \int_{\mathbb{R}}|\widehat{f}(\xi)|d\xi=\int_{\mathbb{R}}\int_{\mathbb{R}}f(x)e^{-2\pi ix\xi}dxd\xi. $$ I do not how to use the condition that $ f $ is continuous at $ 0 $. Can you give me some references or hints?

Cameron L. Williams
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Luis Yanka Annalisc
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  • This can be found in Chandrasekharan K.: Classical Fourier Transforms, Springer (1989) on page 47, Corollary (8.7). – Gerd Sep 17 '22 at 15:37

1 Answers1

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$f_n=f \ast n e^{-\pi n^2 x^2}$ is $L^1$ and its Fourier transform is $\hat{f_n}=\hat{f} e^{-\pi y^2/n^2}$ which is $L^1$ as well (as $\hat{f}$ is bounded).

So we can apply the Fourier inversion theorem and get $$f_n(0)=\int_\Bbb{R} \hat{f_n}(y)dy$$

Then use that $\hat{f}\ge 0$ and that $f$ is continuous at $0$ to get that

$$\|\hat{f}\|_{L^1}=\lim_{n\to \infty}\int_\Bbb{R} \hat{f_n}(y)dy=\lim_{n\to \infty} f_n(0)=f(0) < \infty$$

reuns
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