= Tikhonov regularization of spherical far-field continuation
{title2=$g_{\alpha,n}^m=d_n b_n^m/(1+\alpha|d_n|^2)$}
Let $d_n=ki^{n+1}h_n^{(1)}(kR)$ and map the near-field trace to the far field by $A g=(g_n^m/d_n)_{n,m}$. Quadratic <Tikhonov regularization> minimizes $\|Ag-y^\delta\|^2+\alpha\|g\|^2$ and yields the displayed filtered coefficients. Its data-error gain is at most $1/(2\sqrt\alpha)$. Total error also contains bias: $\|g_\alpha^\delta-g^\dagger\|\le\delta/(2\sqrt\alpha)+\|\alpha(A^*A+\alpha I)^{-1}g^\dagger\|$. A convergent parameter rule has $\alpha\to0$ and $\delta/\sqrt\alpha\to0$. Under $g^\dagger=A^*Aw$, $\|w\|\le C$, the bias is at most $C\alpha$; choosing $\alpha=(\delta/(4C))^{2/3}$ minimizes that error bound. Making $\alpha$ indefinitely large removes noise by shrinking the reconstruction to zero, not by recovering the true field.
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