5. Reconstruction formulas and activation functions🔗

The synthesis activation may be a tempered distribution, including ReLU. Weighted Sobolev spaces make the Fourier pairing continuous; we define them and prove their duality before using regularized synthesis. The later inverse Fourier estimates give an absolutely convergent network integral under Sobolev conditions, with an explicit example of filters that are not band pass.

1. 5.1. Weighted Sobolev activation spaces
2. 5.2. Reconstruction with tempered activations
3. 5.3. Inverse Fourier estimates and dual pairing
4. 5.4. Integral representation under Sobolev conditions
5. 5.5. Example of filters for the reconstruction formula