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 $\theta$.

notadmit such an estimator)? $\qquad$ $\endgroup$