Appendix D. The finite-dimensional case and the dilation obstruction🔗

This appendix compares the weighted construction with the finite-dimensional Fourier calculation. In finite dimension the homogeneous Gaussian mixture has a restricted range of exponents. We then identify the frame operator and explain why direct dilation of a Gaussian spectral measure cannot supply the proposed finite coefficient measure.

1. D.1 A Fourier calculation for comparison
2. D.2 The finite-dimensional reference measure
3. D.3 Filtered backprojection
4. D.4 Why the spectral measure cannot be dilated