VARIATIONAL inequalities (Mathematics), FOURIER series, MATHEMATICAL functions, MATHEMATICS, APPROXIMATION theory
Abstract
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]
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]
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]