The kernel polynomial method based on Jacobi polynomials

The kernel polynomial method based on Jacobi polynomials $P_n^{\alpha ,\beta }(x)$ is proposed. The optimal-resolution positivity-preserving kernels and the corresponding damping factors are obtained. The results provide a generalization of the Jackson damping factors for arbitrary Jacobi polynomials. For $\alpha =\pm 1\mathrm{/}2$, $\beta =\pm 1\mathrm{/}2$ (Chebyshev polynomials of the first to fourth kinds), explicit trigonometric expressions for the damping factors are obtained. The resulting algorithm can be easily introduced into existing implementations of the kernel polynomial method.