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 Sⁿ Rⁿ⁺¹, n 1, 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 L_p(Sⁿ), 1 p , to the space H_r H_rL_p(Sⁿ) 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.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 123–129, September, 1992.The author wishes to express his deep gratitude to Academician S. M. Nikol'skii and Professor P. I. Lizorkin for discussion of the results of the present note.