I understand that $X(k) = \sum_{n=0}^{N-1}x_ne^{-i2\pi \frac{k}{N} n}$ and $x(n) = \frac{1}{N}\sum_{k=0}^{N-1}X_k e^{i2\pi \frac{k}{N} n}$ are the discrete Fourier transform and inverse discrete Fourier transform. How do I translate/interpret/apply these formulas in terms of sampling period and frequency. For example, if I sample $11$ times so that $N=11$, and 1.5 seconds is the sampling period. Then from what little I understand, I should be able to write something that resembles sort of like, (ignoring the sine terms), $y(t)=A_1\cos(2\pi ft) + A_2\cos(2\pi(2f)t)+\ldots$ where $y(0),y(1.5),y(2*1.5),\ldots, y(10*1.5)$ take on the values of those $11$ sampled values and $t$ is time in seconds, $f$ is some lowest frequency needed in the representation.
Now somehow using the inverse discrete Fourier transform $x(n)$ I'm suppose to be able to get at $y(n)$. However, everything in $X(k)$ and $x(n)$ is based around integers (i.e. $n$ and $k$ are integers). Whereas $y(t)$ deals with values like $t=1.5$ and $f$ that are not integers. Can someone show the baby steps to get from $x(n)$ to $y(t)$.