  We address the optimal constants in the strong and the weak Stechkin
inequalities, both in their discrete and continuous variants. These
inequalities appear in the characterization of approximation spaces which arise
from sparse approximation or have applications to interpolation theory. An
elementary proof of a constant in the strong discrete Stechkin inequality given
by Bennett is provided, and we improve the constants given by Levin and
Stechkin and by Copson. Finally, the minimal constants in the weak discrete
Stechkin inequalities and both continuous Stechkin inequalities are presented.
