  In this paper we give exact values of the best $n$-term approximation widths
of diagonal operators between $\ell_p(\mathbb{N})$ and $\ell_q(\mathbb{N})$
with $0<p,q\leq \infty$. The result will be applied to obtain the asymptotic
constants of best $n$-term approximation widths of embeddings of function
spaces with mixed smoothness by trigonometric system.
