Compactly supported smooth function with Laplace transform bounded on a cone

Question

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays bounded for $x\in \mathbf{R}$.

Some more thoughts: for example, if we take $\varphi$ to be supported on the interval $[1,2]$, and take $x$ to be some large number, then the term $e^{tx}$ hehaves like $e^x$, which might(?) be able pulled out from the integration to obtain $e^{x}\int_{\mathbf{R}}\varphi(t)e^{itx}dt$, thus in order for the last expression to be bounded, we need $\int_{\mathbf{R}}\varphi(t)e^{itx}dt$ to have exponential decay! This is not possible as if it is true, we know $\varphi$ will be real analytic from the Fourier inverse formula, which will then be identically zero from the fact that it has compact support.

I'm now really hoping that before pulling out the term $e^{tx}$, the term $e^{itx}$ will generate enough cancellation!

Answer 1

I'll add a few details to the answer by Michael Renardy on MathOverflow. The function $$F(z)=\int_{\mathbb R}\varphi(t)e^{itz}\,dz$$ is

1. Entire (holomorphic in $\mathbb C$), with growth bound $|F(z)|\le Ce^{A|z|}$ coming from $|e^{itz}|\le e^{A|z|}$ where $A=\sup\{|t|:\varphi(t)\ne 0\}$.
2. Bounded on the real line, because for real $z$ we have $ |e^{itz}|=1 $.
3. Bounded on the line $\operatorname{Re}z+\operatorname{Im} z=0$, by assumption.

The Phragmén–Lindelöf principle applies to $F$ in the sectors formed by the lines in 2 and 3. Indeed, the exponential growth bound 1 suffices in any sector of opening angle $<\pi$, and we have angles $\pi/4$ and $3\pi/4$. The principle says that $F$ is bounded in each sector; hence, bounded on $\mathbb C$; hence constant by Liouville's theorem. But then $F\equiv 0$, because the Fourier transform tends to $0$ as $z\to\infty$ along the real axis.