Consider the discrete time signal $x[t] = \{24, 8, 12, 16\}$
When we perform a DFT on this we get:
$\{30\, +0 i,6\, -4 i,6\, +0 i,6\, +4 i\}$, which tells us the frequencies of the various sinusoids in the signal. We can thus think of a way to "interpolate" by simply defining the continuous $x(t)$ to be the summation of those sinusoids:
$x(t) = -4 \sin \left(\frac{\pi n}{2}\right)+6 \cos \left(\frac{\pi n}{2}\right)+3 \cos (\pi n) + 15$
The above is a method that wikipedia calls "trigonometric interpolation."
However, there's also the well-known sinc interpolation which can be done in the frequency domain by zero-padding the DFT output. Conceptually, this seems to be multiplying the frequency domain with a rectangular filter to remove its periodic components, such that the inverse gets you back to a continuous, periodic signal.
However, isn't this the same as just doing a trigonometric interpolation on one period of the DFT (since you only use those frequency components?)
When we plot the two side-by-side, however, we see that they do not match up 
What's the reason for this?