Linear interpolation is suboptimal, as you may know. You understand it in time-domain, but let's look into it in frequency-domain. The sampled signal spectrum would be periodic with period $\omega=2\pi$ (f=1).
Ideally, we could use an ideal low pass filter with cutoff frequency at $f_S/2$ Don't forget that, although not in the picture, negative frequencies are defined.
You may recall that an ideal frequency response means a sinc kind of time-domain impulse response. That means we would need infinite filter coefficients: every sample needs to know about every past and future sample. So why not try something simpler?
That would be, for example, linear interpolation. The linear time-domain response you use resembles a triangular signal, which could also be seen as a sinc in frequency-domain
$H(j\omega)=1 / T · (sin (\omega T /2)/ (\omega/2))^2$
(please check Oppenheim's Signals and systems, section 7.2 for good graphs and a really deep and awesome explanation, you won't have trouble to find it in the net. Discrete time signal processing also includes this matter.)
Really inaccurately, this filter would look like:
Thus, our filter is worse than the ideal (obviously), but now we have a finite time-domain response (a.k.a. FIR filter). This is the way that the time-domain operations you were doing relate to frequency-domain.
For more info concerning how to apply the filter, I advise to check the mentioned books. If you define a filter in time-domain (h[n]) you can apply it using convolution, while if defined in frequency-domain (H(jw)) you can compute the output spectrum as the product of the input and the freq. response.