I'm trying to really understand how multi-layer perceptrons work. I want to prove mathematically that MLP's can classify handwritten digits. The only thing I really have is that each perceptron can operate exactly like a logical operand, which obviously can classify things, and, with backpropagation and linear classification, it's obvious that, if a certain pattern exists, it'll activate the correct gates in order to classify correctly, but that is not a mathematical proof.
The approximation theorem says you can approximate anything. But this is kind of meaningless in so far as you can do KNN and get an arbitrary approximation of your training data also.
Proving CNN correctly extract features is, I don't think possible. Or if it is, something involving VC theory is probably the best you can do.