Strong Approximation by Cesro Means with Critical Index in the Hardy Spaces H p (Sd1) (0 > p 1)

Let (Sd1)={x:|x|=1}be a unit sphere of the ddimensional Euclidean space d and let HpHp(Sd1)(0 > p 1) denote the real Hardy space on Sd1.For 0 > p 1 and fHp(Sd1),let E j (f,H p) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space Hp(Sd1).Given a distribution f on Sd1,its Cesro mean of order > 1 is denoted by k(f).For 0 > p 1, it is known that (p):=d1pd2is the critical index for the uniform summability of kin the metric H p. In this paper, the following result is proved: Theorem Let 0>p>1 and =(p):=d1pd2. Then for fHp(Sd1), j=1N1jj(f)fpHpj=1N1jEpj(f,Hp), where A N (f)B N (f) means that theres a positive constant C, independent of N and f, such that C1AN(f)BN(f)CAN(f). In the case d = 2, this result was proved by Belinskii in 1996.