The operation of convolution is computationally expensive to perform in real space for large number of cells/particles (of the order of 70+ millions), and I wanted to avoid using Fourier methods for different reasons: first of all, the regions of space I am interested in are not periodic; second, I want to use the full resolution of the simulation in the high-resolution regions and not having to interpolate on a coarser grid.

\[ \sin(x) = \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!} = x + \frac{x^3}{6} + \frac{x^5}{120} + \dotsb \\ \sin(x) = \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!} = x + \frac{x^3}{6} + \frac{x^5}{120} + \dotsb \]

$$ \begin{aligned} &(n-1)(n-1) \\ &=n^2-2n+1 \end{aligned} $$

This is inline katex $ \langle f g \rangle (\bm x) $.