Decay of non-negative functions with compact Fourier support

Question

Let $f$ be a function on the real line with $\widehat{f}$ supported in the interval $[-1,1]$. Let's denote the space of such functions with $W_0$.

Let $g\ge 0$ denote a rapidly decaying (and say, continuous, if that matters) function on the real line; what I have in mind is something like the absolute value of a Schwartz function.

Question: Given such a function $g$, does there always exist $f\in W_0$ such that $g(x)\le f(x)$ for all $x\in\mathbb{R}$ ? If yes, can we also find $f\in W=W_0\cap \mathcal{S}$ ?

(Here $\mathcal{S}$ denotes Schwartz functions.)

Presumably, for the second part, rapid decay is not sufficient. On the other hand if $g$ is compactly supported, then the answer to the second question seems to be yes, let $f=\widehat{\phi*\phi}$ for an approriately chosen smooth $\phi$ supported in $[-1/2,1/2]$.

Partial answers or helpful suggestions are welcome as well.

Edit: Sorry for confusion over multiple edits.

Answer 1

Edit: The first version of this was a reply to the first version of the question, where $W$ was defined by saying $\hat f$ is supported in $[-1,1]$, without assuming $f\in\mathcal S$. Let's say that space is $W_0$. We give the solution for $W_0$, then indicate how it could be modified for $W$.

Edit${}^2$: If you're wondering why I repeat the definition of $W_0$ here, it's because the OP is still modifying the question. [Sigh...]

For $W_0$ rapid decrease is more than enough; in fact

There exists $f\in W_0$ with $f(t)\ge 1/(1+t^2)$ for all $t$.

First,

There exists $\psi\in W_0$ with $\psi\ge0$ and $\psi(0)>1$.

(Hint: $\hat \psi=\phi*\phi$...)

Now say $\psi\ge1$ on $(-\delta, \delta)$. Let $$I_n=[(n-1)\delta, (n+1)\delta]\quad(n\in\Bbb Z).$$ Choose $a_n$ with $$\sum_{n\in\Bbb Z}a_n<\infty$$and $$a_n\ge\frac1{1+t^2}\quad(t\in I_n).$$Let $$f(t)=\sum_{n\in\Bbb Z} a_n\psi(t-n\delta).$$

More or less the same thing works for $W$, if $g$ is rapidly decreasing:

If $g$ is rapidly decreasing there exists $f\in W$ with $f\ge|g|$.

First, note you can get $\psi\in W$ by taking $\hat\psi=\phi*\phi$ with $\phi$ smooth.

Now there exists a rapidly decreasing sequence (obvious definition left to you) with $$|g(t)|\le a_n\quad(t\in I_n).$$Define $f$ as above. Then $f\in\mathcal S$, since $\hat f$ is smooth with compact support. (The fact that $a_n$ is rapidly decreasing shows that $m$ is smooth, if $$m(x)=\sum a_ne^{in\delta x};$$ hence $\hat f = m\hat\psi$ is smooth.)