I want to prove that if $f \in C^{1}(\mathbb{R})$ is compactly supported then its Fourier transform is integrable. I was able to prove the result for $f \in C^{2}(\mathbb{R})$ and compactly supported. I used the fact that if $f \in C^{2}(\mathbb{R})$, then $\hat{f}$ is bounded by $\frac{c}{1+{|x|}^{2}}$. So it is integrable. I failed to prove it if $f \in C^{1}(\mathbb{R})$
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1$\begingroup$ Does page 4 of the following link help? math.unc.edu/Faculty/met/s14.pdf $\endgroup$– SuvritCommented Apr 21, 2011 at 7:35
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$\begingroup$ No, but page 6 does :), thanks a lot,... I'm trying to see now if this result can be upgraded to $\mathbb{R}^{n}$ $\endgroup$– jessicaCommented Apr 21, 2011 at 9:07
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$\begingroup$ @Suvrit: The link is broke. Could you please write out the author and title of the paper? $\endgroup$– HansCommented Feb 19, 2019 at 0:59
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$\begingroup$ @Suvrit Link broken. $\endgroup$– coudyCommented Feb 25, 2022 at 10:35
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$\begingroup$ @DenisSerre Are you sure about your comment? I am pretty sure that square integrable functions with square integrable generalised derivative have integrable Fourier transforms, as a consequence of the Cauchy-Schwarz inequality. $\endgroup$– coudyCommented Feb 25, 2022 at 15:00
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1 Answer
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Here is a link to the paper about necessary conditions for the integrability of the Fourier transform:
http://www.heldermann-verlag.de/gmj/gmj16/gmj16043.pdf
It is stated in that paper that sufficient conditions for the integrability of the Fourier transform are given in the book
R. M. Trigub and E. S. Bellinsky, Fourier analysis and approximation of functions. Kluwer Academic Publishers, Dordrecht, 2004.
answered Apr 20, 2011 at 19:57
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1$\begingroup$ The link is broke. Could you please write out the author and title of the paper? $\endgroup$– HansCommented Feb 19, 2019 at 0:43
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1$\begingroup$ I think the paper is: Elijah Liflyand, Necessary Conditions for Integrability of the Fourier Transform, Georgian Mathematical Journal, Volume 16 (2009), Number 3, 553–559 $\endgroup$ Commented Nov 20, 2019 at 20:18