The triangular function is defined as follows:
$h_l(x) = \begin{cases}1-|x|,&|x|<1;\\0&\text{otherwise}.\end{cases}$
According to ccrma.stanford.edu:
"If the output of the interpolator is also sampled, this can be modeled by sampling the continuous-time interpolation result, thereby aliasing the $\text{sinc}^2$ frequency response. [...] The Fourier transform of $h_l(nT/L)$ is the same function aliased on a block of size $ f_s=L/T$ Hz."
What I am thinking: If the Fourier transform $\hat{f}$ has bounded support (bandlimited), then $f$ has unbounded support (infinite length). This is a consequence of the uncertainty principle. But the triangular function (i.e. $f$) is $0$ for $|x| > 1$, so $f$ has bounded support (and $\hat{f}$ cannot be bandlimited). The sampling theorem requires $\hat{f}$ to be supported on some interval $[-B, B]$ (bandlimited). Then we cannot apply the sampling theorem, because $\hat{f}$ is not bandlimited. So aliasing will occur because we cannot find a good sampling rate.
Is my interpretation correct? Aliasing will always occur for $h_l$ no matter whether we look at $\text{DTFT}(\text{sample}(h_l))$ or at $\hat{h_l}$ (Fourier transform)? Then why is linear interpolation so popular in image processing (bilinear interpolation) if it has such a bad frequency response?