Asymptotics of the approximation of continuous and differentiable functions by the singular integrals of de la Vallée Poussin

The complete asymptotic developments in powers of $1/n$ are derived for quantities characterizing approximation by singular integrals of de la Vallée Poussin
\begin{gather*} V_n(f;x)=\frac1{\Delta_n}\int_{-\pi}^\pi f(x+t)\cos^{2n}\frac t2\,dt; \\ \Delta_n=\int_{-\pi}^\pi\cos^{2n}\frac t2\,dt \end{gather*}
of the function classes $\operatorname{Lip}\alpha$, $0<\alpha\le1$, $W^{(r)}$, $r\ge1$ an integer.