ISSN:
1420-8903
Keywords:
Keywords. Triangle mean value property, reciprocal triangle, barycenter, incenter.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary. Given any triangle $ \Delta $ , let $ \Delta(d) $ be the d-skeleton of $ \Delta $ for d = 0, 1, 2. The space $ {\cal H}_{\Delta}(d) $ of all continuous functions in $ {\Bbb R}^2 $ satisfying the mean value property with respect to $ \Delta(d) $ is determined explicitly for each d = 0 , 1, 2. We have $ \dim {\cal H}_{\Delta (d)} = 6 $ if the origin is the barycenter of $ \Delta $ (resp. the incenter of $ \Delta' $ ) for d = 0, 2 (resp. for d = 1), where $ \Delta' $ is the reciprocal triangle of $ \Delta $ ; otherwise $ {\cal H}_{\Delta (d)} = 2 $ . Moreover there exists a homogeneous polynomial F d (x) such that $ {\cal H}_{\Delta (d)} $ is generated by F d (x) as a $ {\Bbb R}[\partial] $ -module, F d (x) being determined explicitly.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s000100050078
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