Some properties of operators of Abel–Poisson type

We obtain asymptotic estimates for approximations of certain classes of continuous and differentiable functions by operators
$$ A_{\gamma,r}(f;x)=\frac1\pi\int_{-\pi}^\pi f(x+t)\Bigl(\frac12+\sum_{k=1}^\infty r^{k^\gamma}\operatorname{cor}kt\Bigr)\,dt $$
for $\gamma=1\,\text{and}\,2$.