Summability of orthogonal expansions of several variables |
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Authors: | Zhongkai Li Yuan Xu |
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Affiliation: | a Department of Mathematics, Capital Normal University, Beijing 100037, China;b Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA |
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Abstract: | Summability of spherical h-harmonic expansions with respect to the weight function ∏j=1d |xj|2κj (κj0) on the unit sphere Sd−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(Sd−1) if and only if δ>(d−2)/2+∑j=1d κj−min1jd κj. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and Sd−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏j=1d |xj|2κj(1−|x|2)μ−1/2 on the unit ball Bd and ∏j=1d xjκj−1/2(1−|x|1)μ−1/2 on the simplex Td. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|2μ(1−t2)λ−1/2 on [−1,1] is studied, which is of interest in itself. |
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Keywords: | Spherical h-harmonics Unit sphere Summability |
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