Let $g$ be a continuous function of moderate growth. I want to prove that if $\int_\mathbb R g\cdot f=0$ for all $f$ in a family of (real-analytic, polynomially-decaying) functions described below, then $g=0$.
The family of functions is made up of those $f$ that are the restriction to $\mathbb R$ of a function holomorphic in a fixed strip $|{\rm Im}(z)| and satisfy $f(x+iy)=O\big((1+x^2)^{-t}\big)$. Obviously, $t$ must be sufficiently large to cancel the moderate growth of $g$. (Holomorphy is needed to guarantee that the $f$ have exponentially decaying Fourier transforms, which is necessary for my larger project.)
This type of result is well-known when the $f$ have compact support (e.g., Theorem 1.2.5 in Hormander's "The Analysis of Linear Partial Differential Operators I"), and there are results in similar situations. For example, if $f=O\big((1+|x|)^{-\alpha}\big)$ for $\alpha>1$, and $\int_\mathbb R f=1$, then the family $f_\epsilon(x)=\epsilon^{-1}f(x/\epsilon)$ works for a wide class of $g$ (Theorem 8.15 in Folland's "Real Analysis"). Unfortunately, if $f$ is analytic in a strip, each $f_\epsilon$ will be analytic in a strip that goes to zero with $\epsilon$, and I think I need them to be analytic in a fixed strip (plus, I'm not sure this method will work for moderately-growing $g$).
I've been trying to prove this using the subfamily (as $t\rightarrow\infty$) $f_t(x)={\Gamma(t)\over\sqrt{\pi}\Gamma(t-1/2)}{a^{2t-1}\over (a^2+x^2)^t}$ but I am currently unable to obtain a sufficient bound on $\int_{|x|>R}f_t\cdot g$ to deduce that $\lim_{t\rightarrow\infty}\int_{|x|>R}f_t\cdot g=0$ (assuming that happens!). I get stuck pretty quickly: by moderate growth, $g(x)\le C(a^2+x^2)^N$, for some $C$, $N$, so $\int_{|x|>R}f_t\cdot g\le C\int_{|x|>R}{\Gamma(t)\over\sqrt{\pi}\Gamma(t-1/2)}{a^{2t-1}\over (a^2+x^2)^{t-N}}$ I'd thought I'd be able to calculate this, but my attempts so far have failed...
To summarize, my specific questions are:
(1) Is it false that $\int_\mathbb R f\cdot g=0$ implies $g=0$, for $f$ in the above family?
(2) Is there a reference to a proof of this or a related fact that I might adapt into a proof?
(3) Is there a bound on $\int_{|x|>R}f_t\cdot g$ from which I can deduce that $\lim_{t\rightarrow\infty}\int_{|x|>R}f_t\cdot g=0$?
(3') Can you replace $f_t$ by $F_t(x)=-i(x+i)f_t(x)$?