The Squared Singularity Kernel \(\psi_{\kappa}\in L^2(\mathcal{SO}(3))\) is a nonnegative function depending on a parameter \(\kappa\in(0,1)\) and is defined by its Chebyshev series
\[ \psi_{\kappa}(t) = \sum\limits_{n=0}^{\infty} \hat{f}_n(\kappa) \, \mathcal U_{2n}(t) \].
where the chebychev coefficients follows a 3-term recurrsion
\(\hat{f}_0 = 1\) \(\hat{f}_1 = \frac{1+\kappa^2}{2\kappa}-\frac1{\log\frac{1+\kappa}{1-\kappa}}\) \(\hat{f}_n = \frac{(2n-3)(2n+1)(1+\kappa^2)}{(2n-1)(n-1)2\kappa} \, \hat{f}_{n-1}(\kappa)-\frac{2\kappa(n-2)(2n+1)}{2n-3} \, \hat{f}_{n-2}(\kappa)\).
Syntax
psi = SO3SquareSingularityKernel(0.2)