Abstract
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 S n-1, where δ ≥λ≔ (n-2)/2, the latter being the critical index for Cesáro summability of Fourier--Laplace series on S n-1. The main purpose of this paper is to extend known results from the unit circle S 1to the general sphere S n-1 with n≥3. We prove six theorems. To prove Theorems 1-3, our machinery is based on the equiconvergent operator E δ N (f) of the Cesáro means σδ N (f) on S n-1 introduced by Wang Kunyang for δ>-1. We prove in Theorem 6 that E δ N (f) is also equiconvergent with σδ N (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.
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Brown, G., Dai, F. & Móricz, F. Strong approximation by Fourier--Laplace series on the unit sphere S n-1 . Acta Mathematica Hungarica 102, 91–116 (2004). https://doi.org/10.1023/B:AMHU.0000023210.94136.89
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DOI: https://doi.org/10.1023/B:AMHU.0000023210.94136.89