Neural network frames on the sphere

We construct a multiresolution analysis of the standard Hilbert space on a Euclidean sphere, which can be implemented directly by neural networks. The neural networks may utilize any sufficiently smooth function as an activation function, and their size can be determined in advance. We define frame operators that can analyze data selected at scattered sites, rather than any particular set of points. The number of vanishing moments increases with the order of the frames. In particular, the frames can detect discontinuities in arbitrarily high order derivatives. The neural networks do not require any training in the traditional sense