The Limits of Discrete Gaussians

Sampling breaks the closure

Errors that compound

The Nyquist wave survives

What the fix must look like

A Gaussian pyramid is built by repeatedly blurring with a sampled Gaussian kernel. Why does the non-closure of sampled Gaussians matter after many levels?

Before downsampling an image by discarding every other pixel, a blur is applied. Why must the blur null the alternating wave exactly rather than attenuate it strongly?

Two Gaussian blurs with variances $\sigma_1^2 = 1$ and $\sigma_2^2 = 1$ are cascaded in the continuous world. Write the resulting variance $\sigma_3^2$ as a number.