8.4. Approximation-error summary🔗

The table gives the mean supremum error at widths 4 and 4096, together with the fitted logarithmic slope. The dashed lines in the figures are proportional to N^{-1/2}.

Experiment	Resolution	Error, width 4	Error, width 4096	Slope
1: ReLU	10	0.41	0.011	−0.53
1: ReLU	100	0.37	0.012	−0.50
1: ReLU	1000	0.42	0.011	−0.51
1: ReLU	10000	0.32	0.010	−0.50
2: Gaussian target	10	11.9	0.46	−0.47
2: Gaussian target	100	11.6	0.43	−0.48
2: Gaussian target	1000	12.8	0.36	−0.52
3: Dirichlet layer	64	0.00010	0.000012	−0.31
3: Dirichlet layer	512	0.00014	0.0000036	−0.53
3: Dirichlet layer	4096	0.000089	0.0000029	−0.50

These experiments illustrate approximation rates rather than computational cost, which grows with the input resolution through the inner products. Similar errors across the tested resolutions occur only once discretization error is sufficiently small. Finite test sets, quadrature and importance resampling remain numerical approximations; the analytic theorems provide the infinite-dimensional guarantees.