9.3. Position among related results🔗

The universality statement is of a type already available as an existence theorem: compact-open density of networks with scalar ridge neurons holds on locally convex spaces for every continuous non-polynomial activation, and global density on weighted spaces is also known. What the ridgelet transform adds is the construction of the approximants: an explicit coefficient for every target with a spectral density and every admissible analysis filter, a Plancherel identity with an identified frame operator, and a sampling argument that yields the finite-width rate and applies to the examples. The rate is a Barron-type statement for a restricted class, consistent with the lower bounds for Lipschitz operator classes. The transform differs from the ridgelet transform for Wiener functionals in its atoms, its parameter measure, and its Plancherel identity on a single space; the manuscript's Appendix F records these comparisons.