DFT Frequency domain analysis and interpolation
A while back, I remember some professor saying that for the N point DFT Frequency Domain, the values for k around N/2 (halfway point for the spectral domain - those around π) are considered high frequencies, and the end points (0, N-1) (those around 0, 2π) are considered low frequencies. I am having trouble seeing this mathematically given the definition for computing the coefficients of the DFT. 


You can define the DFT however you like, as long as in effect the same frequencies are covered - remember, $f(x) = e^{jx}$ is $2\pi$-periodic, so that, for example, the sum over $k=0,\ldots, N-1$ is the same as the sum over $k=-\left\lfloor\frac N2\right\rfloor,\ldots,0,\ldots,+\left\lfloor\frac {N-1}2\right\rfloor$.


  The examples I have seen seem to decide to pad the coefficients around the middle, i.e. for N=4 and a desired sampling of 2N, [1 2 3 4] become [1 2 0 0 0 0 3 4]. Is there a reason why the middle is targeted and not the end like [1 2 3 4 0 0 0 0]?


Again, different definitions of the DFT lead to different forms for zero-padding here.