Suppose we know the DFT of a discrete limited sequence, some $X[k],\ k = 0, 1,\dots ,N-1$. How can we calculate the Fourier Transform of the same signal for a random frequency $\Omega$?
EDIT:
How about this, we can get the starting signal with inverse DFT, like: $$x[n] = \frac{1}{N}\sum\limits_{k=0}^{N-1} X[k]e^{j\frac{2\pi}{N}kn}$$ and then get the Fourier Transform normally with: $$\begin{align} X(e^{j\Omega}) &= \sum\limits_{n = -\infty}^{+\infty} x[n]e^{-j\Omega n}\\ &= \frac{1}{N}\sum\limits_{n = -\infty}^{+\infty} e^{-j\Omega n}\sum\limits_{k=0}^{N-1}X[k]e^{j\frac{2\pi}{N}kn} \end{align}$$
Is this correct?
EDIT2: As @Matt L. correctly states, I want the Discrete-time Fourier transform rather than the normal Fourier Transform, of course. In my native-language textbook Fourier Transform means both FT for continuous signals and DTFT for discrete signals. I hoped that that was clear from the edit. :)