Let $L:\mathbb{R}^2 \rightarrow \mathbb{R}$ be the Laplacian of the 2D Gaussian, $$L(\boldsymbol{x}) = \frac{1}{2\pi\sigma^4}\left( \frac{\left\|\boldsymbol{x}\right\|^2}{\sigma^2}-2\right)\exp\left(-\frac{\left\|\boldsymbol{x}\right\|^2}{2\sigma^2} \right)$$ and~$d:\mathbb{R}^2 \rightarrow \mathbb{R}$ a signed distance function to some simple closed contour in the plane (https://en.wikipedia.org/wiki/Signed_distance_function). It is continuous and verifies $$\left\| \nabla d(\boldsymbol{x}) \right\|=1$$ almost everywhere (the signed distance has ridges, where $\nabla d$ is undefined). I'm interested in the convolution between $d$ and $L$ at some point $\boldsymbol{x}_0$, $$(d*L)(\boldsymbol{x}_0) = \int_{\mathbb{R}^2} d(\boldsymbol{x})L(\boldsymbol{x}_0-\boldsymbol{x})d\boldsymbol{x}$$ No closed-form expression can be usually derived. For now, I've tackled the problem by considering the 2nd order Taylor approximation of $d$ near $\boldsymbol{x}_0$, $$\tilde{d}(\boldsymbol{x}) = (\boldsymbol{x}-\boldsymbol{x}_0)^\mathrm{T} \nabla d(\boldsymbol{x}_0) + (\boldsymbol{x}-\boldsymbol{x}_0)^\mathrm{T} \nabla^2 d(\boldsymbol{x}_0) (\boldsymbol{x}-\boldsymbol{x}_0),$$ (where $\nabla^2 d$ is the Hessian matrix) We have $$d(\boldsymbol{x}) = \tilde{d}(\boldsymbol{x}) + O\left(\left\|\boldsymbol{x}-\boldsymbol{x}_0\right\|^3 \right)$$
I've replaced $d$ by its approximation $\tilde{d}$ and was able to derive interesting closed-form expressions for $(\tilde{d}*L)(\boldsymbol{x}_0)$. The problem is that I am unable to get a handle on the error term on $(\tilde{d}*L)(\boldsymbol{x}_0)$ with respect to $(d*L)(\boldsymbol{x}_0)$ (as $\left\|\boldsymbol{x}-\boldsymbol{x}_0\right\|$ grows, one can easily appreciate that $L$ tends to $0$ mush faster than $d$ grows, so the error $O\left(\left\|\boldsymbol{x}-\boldsymbol{x}_0\right\|^3 \right)$ gets quickly canceled, but that's all I can tell)
Is there a way to determine the order of the error term ? i.e. to find something like $$(d*L)(\boldsymbol{x}_0) = (\tilde{d}*L)(\boldsymbol{x}_0) + O(?)$$