The approximation of a Hölder class of two variables by Riesz spherical means

For periodic functions of the Hölder class $H_2^\alpha$ ($0<\alpha\le1$) defined in the two-dimensional space $E_2$, we find the asymptotic form as $R\to+\infty$ of the quantity
$$\sup_{f\in H_2^\alpha}\|S_r^\delta(x,f)-f(x)\|_{C(E_2)}\left(\delta>\frac12+\alpha\right),$$
where $S_R^\delta(x,f)$ is the Riesz spherical mean of order $\delta$ of the Fourier series of the function $f(x)$.