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improved formatting and grammar. Added a top level tag.
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Jeremy Rouse
Jeremy Rouse
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If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how can I show that $\hat{f}$ cannot satisfy $\hat{f}(x)=O(e^{-\epsilon|x|})$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ functionfunctions with compact support.

If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how can I show that $\hat{f}$ cannot satisfy $\hat{f}(x)=O(e^{-\epsilon|x|})$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ function with compact support.

If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how can I show that $\hat{f}$ cannot satisfy $\hat{f}(x)=O(e^{-\epsilon|x|})$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ functions with compact support.

improved formatting and grammar
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edit approved Jun 27, 2014 at 17:19
avpaderno
edit approved Jun 27, 2014 at 17:19
avpaderno
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If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how tocan I show that $\hat{f}$ cannot satisfy $$ \hat{f}(x)=O(e^{-\epsilon|x|}) $$ for $\hat{f}(x)=O(e^{-\epsilon|x|})$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ function with compact support.

If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how to show that $\hat{f}$ cannot satisfy $$ \hat{f}(x)=O(e^{-\epsilon|x|}) $$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ function with compact support.

If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how can I show that $\hat{f}$ cannot satisfy $\hat{f}(x)=O(e^{-\epsilon|x|})$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ function with compact support.

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Tomas
Tomas
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Paley-Wiener type theorem for integral functions with compact support

If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how to show that $\hat{f}$ cannot satisfy $$ \hat{f}(x)=O(e^{-\epsilon|x|}) $$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ function with compact support.