The spherical modal DtN map is well-defined and exact for smooth data

By [F3], $\mathrm{\mid }{\tau }_{\mathrm{\ell }}\mathrm{\mid }\le k+(\mathrm{\ell }+1)\mathrm{/}R$. The $C^p$ estimate in [F5] and the super-polynomial decay in [F2] show that for every $p$, ${\sum }_{\mathrm{\ell }}\mathrm{\parallel }{\tau }_{\mathrm{\ell }}{g}_{\mathrm{\ell }}{\mathrm{\parallel }}_{C^p}<\mathrm{\infty }$. Thus the defining series converges in every smooth norm to a smooth function $T_{k,R}g$.

For $r\ge R$, the positive coefficient formula for $\mathrm{\mid }{h}_{\mathrm{\ell }}(z){\mathrm{\mid }}^2$ in [F3] makes $\mathrm{\mid }{h}_{\mathrm{\ell }}(kr)\mathrm{/}{h}_{\mathrm{\ell }}(kR)\mathrm{\mid }\le 1$. The derivative estimate in [F3] bounds one radial derivative of the ratio by $k+(\mathrm{\ell }+1)\mathrm{/}r$ times this ratio; repeated radial derivatives are bounded by polynomials in $1+\mathrm{\ell }$ on each compact annulus by differentiating the radial ODE in [F5]. Angular derivatives likewise contribute only polynomial factors. Thus [F2] and the evaluation bound make the extension series and its derivatives through order two uniformly convergent on $R\le r\le R_1$. Termwise differentiation is valid; each summand solves the Helmholtz equation by [F5], its trace is $g_{\mathrm{\ell }}$, and its radial derivative at $R$ is ${\tau }_{\mathrm{\ell }}{g}_{\mathrm{\ell }}$. Summing proves the trace, derivative, and PDE claims.

For each fixed $\mathrm{\ell }$, the finite formula in [F3] gives $r({\mathrm{\partial }}_r-ik)(h_{\mathrm{\ell }}(kr)\mathrm{/}{h}_{\mathrm{\ell }}(kR))\to 0$. The last bound in [F3], multiplied by $\mathrm{\parallel }{g}_{\mathrm{\ell }}{\mathrm{\parallel }}_{\mathrm{\infty }}$, dominates the absolute value of this degree's radiation residual by $(\mathrm{\ell }+1)e^{2kR}\sqrt{(2\mathrm{\ell }+1)\mathrm{/}(4\pi )}\mathrm{\parallel }{g}_{\mathrm{\ell }}{\mathrm{\parallel }}_2$, a summable sequence by [F2]. Dominated convergence for this series proves uniform convergence of the full radiation residual to zero.

Let $w$ be any other classical outgoing extension of $g$. By [F4], its degree-$\mathrm{\ell }$ projection is $d_{\mathrm{\ell }}{h}_{\mathrm{\ell }}^{(1)}(kr)$ for a fixed $d_{\mathrm{\ell }}\in {\mathcal{H}}_{\mathrm{\ell }}$. Its trace at $R$ is $g_{\mathrm{\ell }}$, so $d_{\mathrm{\ell }}{h}_{\mathrm{\ell }}^{(1)}(kR)=g_{\mathrm{\ell }}$. Nonvanishing in [F3] forces $d_{\mathrm{\ell }}=g_{\mathrm{\ell }}\mathrm{/}{h}_{\mathrm{\ell }}^{(1)}(kR)$, the same coefficient as the constructed extension in [F1]. Their difference has every harmonic projection zero and is continuous; density in [F2] implies the difference vanishes. This proves uniqueness and the well-definedness claimed in [F1].