# Frequency response and sampling theorem for triangular function

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?

• Frequency response is not always important or even relevant in image processing. Note that most image processing operations are non-linear and therefore cannot even be analyzed using Fourier theory, let alone preserve band-limitness. Jul 12, 2020 at 20:24