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Consider the Gaussian $G(x):=e^{-x^2}$ on the real line, and localize it to the region $|x|\sim 2^k$ by multiplying it by an appropriate smooth cut-off. More precisely, take $\phi\in C_0^\infty(\mathbb{R})$ supported in the region $\left \{x\in\mathbb{R}: \frac{1}{2}<|x|\leq2 \right\}$ such that $0\leq\phi\leq 1,$ and consider $$G_k(x):=\phi(2^{-k}x)G(x).$$ It is straightforward to check that $\|G_k\|_{L^1}\lesssim 2^ke^{-4^k}$. My question is: what can be said about $\|\widehat{G_k}\|_{L^1}$? In particular, what decay (if any) do you get in terms of $k$?

Thank you.

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    Not to detract from the question being asked which is of interest in its own right, but most people take $(1/\sqrt{2\pi})\exp(-x^2/2)$ to be the standard Gaussian (density) on the real line.2011-10-12
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    @DJC: I would think that your comment contains enough information to post it is an answer.2011-10-12
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    @Dilip: Edited, thanks!2011-10-13

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