Let $f\in L^1(\mathbb{R})$ such that $$\int_0^\infty e^{-yt}e^{ixt}\widehat{f}(t)dt=0,……(i)$$ $$\int_{-\infty}^0 e^{yt}e^{ixt}\widehat{f}(t)dt=0, ……(ii)$$ for all $x\in \mathbb{R}, y>0.$

Questions:

1. Is it true that $f=0?$
2. If satisfies only condition $(i)$, then can we conclude that $\widehat{f}$ vanishes identically on $[0,\infty)?$

Note: I can prove 2.(hence 1.) assuming the extra condition that $\widehat{f}\in L^1(\mathbb{R})$.

Let $f\in L^1(\mathbb{R})$ such that $$\int_0^\infty e^{-yt}e^{ixt}\widehat{f}(t)dt=0,……(i)$$ $$\int_{-\infty}^0 e^{yt}e^{ixt}\widehat{f}(t)dt=0, ……(ii)$$ for all $x\in \mathbb{R}, y>0.$

Questions:

1. Is it true that $f=0?$
2. If $f$ satisfies only condition $(i)$, then can we conclude that $\widehat{f}$ vanishes identically on $[0,\infty)?$

Note: I can prove 2.(hence 1.) assuming the extra condition that $\widehat{f}\in L^1(\mathbb{R})$.