Transform Pairs and DFT Properties

A pocket table of pairs

Energy is conserved

Shifts become phase ramps

Modulation splits the spectrum

A box signal of half-width $L$ is transformed. Why does the peak of its spectrum sit at the origin with height $2L+1$?

An image is translated by several pixels and both versions are transformed. Comparing the two spectra, what is guaranteed to be identical?

A box signal has half-width $L = 2$, so its nonzero run holds samples from $-2$ to $2$. Write the height of its DFT peak at the origin.