2. The Gaussian-weighted ridgelet transform
This chapter is Section 3 of the manuscript together with Appendix A (the Gaussian mixture),
Appendix G (the finite-dimensional case and the dilation obstruction), Appendix H (abstract
weights), and Appendix I (explicit filters). The input measure is the centred Gaussian
\mu_Q=\mathcal N(0,Q), the direction measure is the homogeneous Gaussian mixture
\nu_\alpha, and the parameter measure is \lambda_\alpha=\nu_\alpha\otimes\mathrm dc.
The formalization states the core theory for an abstract pair (\mu,\nu), as in Appendix H:
\mu a Borel probability measure on H and \nu a \sigma-finite Borel measure with
full support that is homogeneous of degree \alpha. The Gaussian pair
(\mu_Q,\nu_\alpha) is the instance. The Hilbert space \mathcal E_\alpha is represented by
the closed subspace \mathcal K_\alpha=\overline{\mathcal G_Q(\mathcal D_\alpha)}\subseteq
L^2(\nu_\alpha), to which it is unitarily equivalent by Lemma 2.3.2.