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Almansi-type theorems in Clifford analysis

✍ Scribed by Helmuth R. Malonek; Guangbin Ren

Publisher
John Wiley and Sons
Year
2002
Tongue
English
Weight
115 KB
Volume
25
Category
Article
ISSN
0170-4214

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✦ Synopsis

Abstract

In this paper, we consider functions defined in a star‐ike omain Ω⊂ℝ^n^ with values in the Clifford lgebra C__𝓁~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.

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