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Almansi‐type theorems in Clifford analysis
Author(s) -
Malonek Helmuth R.,
Ren Guangbin
Publication year - 2002
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.387
Subject(s) - clifford analysis , mathematics , dirac operator , dirac (video compression format) , kernel (algebra) , type (biology) , clifford algebra , function (biology) , star (game theory) , pure mathematics , operator (biology) , combinatorics , decomposition , mathematical physics , algebra over a field , mathematical analysis , quantum mechanics , physics , ecology , biochemistry , chemistry , repressor , evolutionary biology , gene , transcription factor , neutrino , biology
Abstract In this paper, we consider functions defined in a star‐ike omain Ω⊂ℝ n with values in the Clifford lgebra C 0, n which are polymonogenic with respect to the (left) Dirac operator D = ∑ j =1 n e j ∂ / ∂x j , i.e. they belong to the kernel of D k . We prove that any polymonogenic function f has a ecomposition of the form f = f 1 + xf 2 +···+ x k −1 f k, where x = x 1 e 1 +···+ x n e n and fj , j =1,…, k , are monogenic functions. This generalizes classical Almansi theorem for polyharmonic functions as well e Fischer decomposition of polynomials. Similar results tained for the powers of weighted Dirac operators of the form D̃ =∣ x ∣ −α xD , α ∈ℝ\{0}. Copyright © John Wiley & Sons, Ltd.