\frac{k^2}{2\,r_D^2}+\ldots\right\}\right)\\
&\qquad \qquad -\frac{4(\hat{J}_\phi)_k}{\mathfrak{D}(2\nu-2)} \times\frac{\Gamma(-\nu)}{\Gamma(\nu+2)}\left(\frac{k}{2}\right)^{2\nu}+\ldots  \\
\end{split}$$ where the momentum dependent coefficient ${\mathfrak D}$ is $$\begin{split}
\mathfrak{D}&\equiv 1-\frac{4(2\nu-1)}{(2\nu)^2(2\nu-2)} \frac{k^2}{2\,r_D^2}+
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2\,r_D}\right)^{2\nu}\left\{1-\frac{1}{\nu^2(\nu-1)} \right.\\
&\left.\qquad \qquad \qquad+\; \frac{(2 \nu +1)^4-4 (2 \nu +2)^2+7}{(2 \nu)^2
   (2 \nu +2)^3}\;\frac{k^2}{r_D^2}+\ldots\right\}\\
\end{split}$$ As we saw in the $k=0$ case, we have yet again determined a state dependent source on the boundary for the primary operator ${\cal O}_\Phi$. The key feature to note from the above analysis, is that the expression for the boundary source $J_\phi$ is non-analytic in $k$ and hence non-local when Fourier-transformed back to position space. Hence, we see that in general we have a map between the non-local double trace deformation on the boundary and the Dirichlet data on $\Sigma_D$ (similar non-local double-trace deformations were explored earlier in [@Marolf:2007in]).

A general proposal for linear systems
-------------------------------------

From the analysis of the free scalar wave equation in the picture is rather clear. In the CFT, in general one can make the source a non-local functional of the vev of the primary operator. Usually such a function can be fed into the holographic dictionary via a ‘state-dependent’ boundary condition, which whilst somewhat unnatural from a field theory is a perfectly sensible boundary condition to consider. For 