/*=============================================================================
Copyright (c) 2001-2008 Joel de Guzman
Copyright (c) 2001-2008 Hartmut Kaiser
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at
=============================================================================*/
#ifndef BOOST_SPIRIT_INCLUDE_CLASSIC_STORED_RULE
#define BOOST_SPIRIT_INCLUDE_CLASSIC_STORED_RULE
#include
#endif
Q:
Which are the minimal aires units of the space $L^2(\mathbb R^n)$?
If $f\in L^2(\mathbb R^n)$, $n>2$, then can we say that
$$
\int_{\mathbb R^n}\vert f(x)\vert^2\,dx\geq\frac{1}{n^2}\int_{\mathbb R^n}\vert\hat f(\xi)\vert^2\,d\xi.
$$
There are a lot of constants, so you better compute it.
My thought is that if the rhs is large, the lhs is not less than than the rhs (by the Riemann-Lebesgue lemma), but that’s a guess.
A:
The answer is yes.
Take $g(x)=e^{ -x^2}$, then $g$ is in $L^2(\Bbb R^1)$, and $g
otin L^1(\Bbb R^1)$. Also, $\hat g(\xi)=e^{ -\xi^2}$.
Now, $||g||_2=||g||_1$. Indeed, $||g||_1\le \int |g|\le||g||_2$, which implies that $||g||_1=||g||_2$ by the RHS is dominated by the LHS. Also, $\int |\hat g(\xi)|^2=\
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