Best approximation and de la Vallée–Poussin sums

For the class $C_\varepsilon=\{f\in C_{2\pi}: E_n[f]\leqslant\varepsilon_n, n\leqslant\mathbf{Z}_+\}$, where $\{\varepsilon_n\}_{n\in\mathbf{Z}_+}$ is a sequence of numbers tending monotonically to zero, we establish the following precise (in the sense of order) bounds for the error of approximation by de la Vallée–Poussin sums:
$$ c_1\sum_{j=n}^{2(n+l)}\frac{\varepsilon_j}{l+j-n+1}\leqslant\sup_{f\in C_\varepsilon}||f-V_{n,l}(f)||_C \leqslant c_2\sum_{j=n}^{2(n+l)}\frac{\varepsilon_j}{l+j-n+1}\qquad(n\in\mathrm{N}),\eqno{(1)} $$
where $c_1$ and $c_2$ are constants which do not depend on $n$ or $l$. This solves the problem posed by S. B. Stechkin at the Conference on Approximation Theory (Bonn, 1976) and permits a unified treatment of many earlier results obtained only for special classes $C_\varepsilon$ of (differentiable) functions. The result (1) substantially refines the estimate (see [1])
$$ ||V_{n,l}(f)-f||_C=O(\log n/(l+1)+1)E_n[f]\qquad(n\to\infty)\eqno{(2)} $$
and includes as particular cases the estimates of approximations by Fejér sums (see [2]) and by Fourier sums (see [3]).