The question is this: Does there exist an integrable function $f\colon\mathbb R\to\mathbb R$ such that $f$ differs from $0$ on a set of nonzero Lebesgue measure and 
\begin{equation}
 \int\_{\mathbb R}g\_\theta(x)f(x)\,dx=0 \tag{1}
\end{equation}
for all real $\theta$, where $g\_\theta(x):=e^{-(\theta x-1)^2/2}$? 

More generally, one may ask this: Does there exist a nonzero tempered distribution $F$ on $\mathbb R$ such that $\langle g\_\theta,F\rangle=0$ for all real $\theta$? The latter question can of course be restated as follows: Does there exist a nonzero tempered distribution $\hat F$ on $\mathbb R$ such that $\langle h\_\tau,\hat F\rangle=0$ for all real $\tau$, where $h\_\tau(t):=e^{i\tau t-\tau^2 t^2/2}$? 

One may note that $(1)$ will hold for all $\theta>0$ if $f=I\_{(0,\infty)}-c$, where $I\_{(0,\infty)}$ is the indicator function of the set $(0,\infty)$ and $c=\frac1{\sqrt{2\pi}}\int\_0^\infty e^{-(u-1)^2/2}\,du$. However, the question is whether $(1)$ can hold for (an essentially nonzero $f$ and) all real $\theta$.