Our work is based on a recently introduced mathematical theory of deep convolutional neural networks (DCNNs).
It was shown that DCNNs are stable with respect to deformations of bandlimited input functions.
In the present paper, we generalize this result: We prove deformation stability on Sobolev spaces.
Further, we show a weak form of deformation stability for the whole input space L2.
The basic components of DCNNs are semi-discrete frames.
For practical applications, a concrete choice is necessary.
Therefore, we conclude our work by suggesting a construction method for semi-discrete frames based on bounded uniform partitions of unity (BUPUs) and give a specific example that uses B-splines.