Given an finite interval $(a,b)$, what is the guarantee for existence of a $L_2$ function on $\mathbb{R}$ such that whose $L_2$ Fourier transform has support in $(a,b)$?
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Try to calculate the inverse Fourier transform of something very simple like a hat function. – Ian Dec 10 '15 at 14:42
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Let $\phi$ be the inverse Fourier function of the characteristic function of $[a,b]$: $$ \phi(x)=\frac{1}{\sqrt{2\,\pi}}\int_a^be^{-ix\xi}\,d\xi=\frac{i}{\sqrt{2\,\pi}}\frac{e^{-ibx}-e^{-iax}}{x}. $$ $\phi$ is in $L^p$for all $p>1$, and is an example. You can construct more as follows: for any $f\in L^1\cap L^2$, $\phi\ast f$ is in $L^2$ and $\widehat{\phi\ast f}=\hat\phi\,\hat f$ has support in $[a,b]$.
Julián Aguirre
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