Maps, theory discussion

[Kamog]

In the summation equations, what are the terms x, y, N, m, n, and l?
The first equation looks like the proof for the commutative property of a convolution between x and y, i.e. if you convolute x with y, it gives the same result as y convoluted with x. How does this relate to the quadrant maps?

And what do fourier theorms and bandwidths have to do with the maps. Why are we using a continuous integral from minus infinity to plus infinity, when the quantity we are discussing is not continuous but discrete. The summation should suffice, shouldn't it?

Nerds are tasty but they're bad for your teeth. I think they're just globs of sugar with a bit of color and artificial flavor.