Let $v(x)\in L^1\cap L^2$ be a function, and $w(y)\in L^2$ be its Fourier transform. Let $\{w_k\}$ be an infinite sequence of samples from $w(y)$, sampled $T$ apart. Now,
$$v(x)=\sum_{k=-\infty}^{\infty}w_k e^{ikx}$$
in the $L^2$ norm sense. Is it possible to write $w_k=\displaystyle\int_0^{2\pi}v(x)e^{-ikx}dx$, since $v(x)$ is Riemann integrable, or should I use the inverse discrete transform?
The distance between the sampling points $T$ should be 1 for your equations to hold.
Assuming everything to be nice the discrete back transform goes over to a Riemann integral as $n\to\infty$. So you can use your formula to calculate the back transform...
As always: functions where everything goes well lie dense and the resulting $L^2$ norms are bounded by Plancherel such that this argument can be extended to arbitrary functions in $L^2$.