On approximation of functions on the sphere

Let $S^n$ be the unit sphere in $\mathbf R^{n+1}$ ($n\geqslant 1$) with center at the origin of coordinates, and let $\|*\|_p$ be the norm in the space $L_p(S^n)$, $1\leqslant p\leqslant\infty$ $(L_\infty(S^n)\equiv C(S^n))$. Problems posed by Butzer, Johnen [4], and Wehrens (Approximationstheorie auf der Einheitskugel in $R^3$. Legendre-Transformationsmethoden und Anwendungen, Forschungsberichte Landes Nordrhein-Westfalen No. 3090, 1981) are solved; namely, a direct theorem on best approximation is proved for the modulus of smoothness of arbitrary (fractional) order $r$ $(r>0)$
$$ \omega_r(f;\tau)_p\colon=\sup_{0<t\leqslant\tau}\Big\|(E-\operatorname{sh}_t)^{r/2}f\Big\|_p,\qquad 0<\tau<\pi, $$
where $\operatorname{sh}_t$ is the shift operator on the sphere,
$$ (\operatorname{sh}_tf)(\Theta)=\frac{\Gamma (n/2)}{2\pi^{n/2}(\sin t)^{n-1}}\int_{\Theta\cdot \mu=\cos t}f(\mu)\,dt(\mu),\qquad 0<t<\pi, $$
and its equivalence to the $K$-functional is proved. Special cases of the results established were known from work of Kushnirenko, Butzer, and Johnen, Lofstrom and Peetre, Pawelke, Lizorkin and Nikol'skii, Kalyabin, and others.