Let $f$ be a function on $\Bbb R$ that satisfies
$$f(x) = O(e^{-\pi x^2})\quad\text{and}\quad\hat{f}(\xi) = O(e^{-\pi\xi^2}).$$
Then if $f$ is even then $\hat{f}$ extends to an even entire function. Moreover, if $g(z) = \hat{f}(z^{1/2})$, then g satisfies
$$|g(x)|\leq ce^{-\pi x}\quad\text{and}\...
Let $f$ be a function on $\Bbb R$ that satisfies
$$f(x) = O(e^{-\pi x^2})\quad\text{and}\quad\hat{f}(\xi) = O(e^{-\pi\xi^2}).$$
Then if $f$ is even then $\hat{f}$ extends to an even entire function. Moreover, if $g(z) = \hat{f}(z^{1/2})$, then g satisfies
$$|g(x)|\leq ce^{-\pi x}\quad\text{and}\...
I am reading through a paper that uses a geometric construction not quite clear to me.
Namely, let $B$ be a closed ball in $\mathbb R^n$ centered at $x$.
Let $s \in B$.
Next, let $w$ be an arbitrary vector.
Then, for a sufficiently small $\varepsilon > 0$, there is $z \in \partial B$ such that
$$...
