The kernel polynomial method based on Jacobi polynomials

TL;DR

This paper introduces a generalized kernel polynomial method using Jacobi polynomials, deriving optimal damping factors and providing explicit formulas for special cases, enhancing existing spectral approximation techniques.

Contribution

It generalizes Jackson damping factors for Jacobi polynomials and offers explicit formulas for Chebyshev cases, improving the kernel polynomial method.

Findings

Derived optimal positivity-preserving kernels and damping factors.

Generalized Jackson damping factors for arbitrary Jacobi polynomials.

Provided explicit trigonometric expressions for Chebyshev polynomials.

Abstract

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.