Exact Jackson–Stechkin inequality in the space $L_2$ on hyperboloid

The best mean square approximation of an arbitrary function on the hyperboloid $\mathbb H\subset\mathbb R^3$ by elements of the subspace of functions with the bounded spectrum in the sense of Mehler–Fock is estimated, from above by the $r$th $(r\geq 1)$ modulus of continuity generated by generalized shift (connected with the hyperboloid). The constant in the inequality is exact. Estimates for the least value of the argument of the modulus of continuity are also found when the exact Jackson–Stechkin constant is minimal.