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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

We study the strong approximation properties of the Cesáro means of order δ of the Fourier-Laplace expansion of functions integrable on the unit sphere Sn-1, where δ &gT; λ := (n - 2)/2, the latter being the critical index for Cesáo summability of Fourier-Laplace series on Sn-1. The main purpose of this paper is to extend known results from the unit circle S1 to the general sphere Sn-1 with n &gT; 3. We prove six theorems. To prove Theorems 1–3, our machinery is based on the equieonvergent operator ENδ(f) of the Cesáro means σNδ (f) on Sn-1 introduced by Wang Kunyang for δ > -1. We prove in Theorem 6 that ENδ(f) is also equiconvergent with σNδ(f) for δ > 0 in the case of strong approximation. To prove Theorems 4 and 5, we rely on known equivalence relations between K-functionals and moduli of continuity. [ABSTRACT FROM AUTHOR]

In this paper, we state and prove some new inequalities related to the rate of Lp approximation by Cesàro means of the quadratic partial sums of double Vilenkin-Fourier series of functions from Lp. [ABSTRACT FROM AUTHOR]