  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 $\alpha =\pm 1/2$, $\beta =\pm 1/2$
(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.
