I got confused with the following question when studying Fourier analysis:
If there's a function $f\in L^2(\mathbb{R})$, with the Riemann integral $\int f(t) e^{-2 \pi it \gamma} dt \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ is it true that $\int f(t) e^{-2 \pi it \gamma} dt$ is in fact the Fourier transform of $f$ defined rigorously in $L^2$ space? (i.e. the limit of a sequence of Fourier transform of functions in $L^1(\mathbb{R}) \cap L^2(\mathbb{R}).$) Thanks for any help.