Equivalence of $K$-functional and modulus of smoothness of functions on the sphere

In the present note certain fundamental estimates of the constructive theory of functions on the sphere $S^n\subset\mathbf{R}^{n+1}$, $n\geqslant1$, are sharpened on the basis of the equivalence of the $K$-functional and the modulus of smoothness of functions. In particular a Bernshtein-type inequality for spherical polynomials is made more precise. The estimates obtained are applied to deduce a membership criterion for the function f $f\in L_p(S^n)$, $1\leqslant p\leqslant\infty$, to the space $H_r^{\omega}L_p(S^n)$ depending on the growth of the norm of derivatives of best approximation polynomials of the function $f$, which is an analog of a result found by S. B. Stechkin related to continuous periodic functions.