Before revision: We analyze the popular kernel polynomial method (KPM) for approximating the spectral density (eigenvalue distribution) of a real symmetric (or Hermitian ) matrix A\in \mathbb{R^{n\times n}}  . We prove that a simple and practical variant of the KPM algorithm can approximate the spectral density to \epsilon accuracy in the Wasserstein-1 distance with roughly O({1}/{\epsilon}) matrix-vector multiplications with A.
After revision: We analyze the popular kernel polynomial method (KPM) for approximating the spectral density (eigenvalue distribution) of a real symmetric (or Hermitian ) matrix A\in \mathbb{R^{n\times n}} an n\times n Hermitian matrix A . We prove that a simple and practical variant of the KPM algorithm can approximate the spectral density to \epsilon accuracy in the Wasserstein-1 distance with roughly O({1}/{\epsilon}) matrix-vector multiplications with A.