4. Representation and reconstruction🔗

A spectral density determines explicit network coefficients. We first establish the spectral-target and coefficient-decay estimates, then prove the integral representation. For reconstruction in \mathcal E_\alpha, we next identify weak synthesis with the integral network and prove Hermite recovery. The frame operator and vector-valued extension follow from these preceding ingredients.

1. 4.1. Spectral targets and coefficient estimates
2. 4.2. Integral representation
3. 4.3. Synthesis, Hermite recovery, and reconstruction
4. 4.4. Vector-valued targets