Strong Approximation by Cesàro Means with Critical Index in the Hardy SpacesHp(0<pࣘ 1).

Letbe a unit sphere of thed-dimensional Euclidean space Rd and let(0 < p= 1) denote the real Hardy space onFor 0 < p= 1 andletEj(f,Hp) (j= 0, 1, ...) be the best approximation offby spherical polynomials of degree less than or equal toj, in the spaceGiven a distributionfonits Cesàro mean of order d>-1 is denoted byFor 0<p= 1, it is known thatis the critical index for the uniform summability ofin the metricHp. In this paper, the following result is proved:TheoremLet0<p<1andThen forwhereAN(f)˜BN(f)means that there’s a positive constant C, independent of N and f, such thatIn the cased= 2,this result was proved by Belinskii in 1996. [ABSTRACT FROM AUTHOR]