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Definiteness of the Peano Kernel Associated with the Polyharmonic Mean Value Property
Author(s) -
Haußmann Werner,
Kounchev Ognyan
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001034
Subject(s) - mathematics , positive definiteness , property (philosophy) , definiteness , kernel (algebra) , pure mathematics , corollary , ball (mathematics) , mean value , mathematical analysis , positive definite matrix , statistics , physics , philosophy , eigenvalues and eigenvectors , linguistics , epistemology , quantum mechanics
The definiteness of the Peano kernel is proved for a functional associated with the mean‐value property of Picone and Bramble and Payne for polyharmonic functions in the ball. An important corollary of this is that if a function f satisfying (−1) p Δ p f >0 vanishes on p concentric spheres centered at 0, then f (0)>0. This generalizes a well‐known property of subharmonic functions (which arise in the special case p = 1).