8.2. Experiment 2: synthesis of the weighted Gaussian target🔗

Use Example 7.2 with \alpha=1, P=Q=\operatorname{diag}(j^{-2}), W=I, and \rho(t)=\mathrm{He}_4(t)e^{-t^2/2}. Its Fourier transform is \sqrt{2\pi}\,\omega^4e^{-\omega^2/2}, and its admissibility constant is 6. Although this filter is not band pass, the analyzed-target identity only needs admissibility. Finite variation and the second parameter moment follow directly from the explicit coefficient and the Gaussian decay estimate.

The target 6f_g, representing 6T_\alpha f_W, is synthesized from its explicit coefficient measure. Appendix E describes the quadrature and importance-resampling procedures used to evaluate this target and draw its parameters.

For d=10,100,1000 and 100 test inputs drawn by the rule of Experiment 1, ten trials compare f_N=(V/N)\sum_j\operatorname{sgn}(\gamma_g(\theta_j))\rho(\langle a_j,\cdot\rangle-b_j) with 6f_g, whose supremum on the test set is about 12.5. The error again decays approximately as N^{-1/2} across the tested resolutions.

The direction law is approximated by a weighted proposal pool shared across trials at each resolution. The reported trial variation is conditional on that pool and does not measure its approximation error; Appendix E describes this limitation and the sampling procedure.