Abstract
AbstractIn this paper, we consider functions defined in a star‐ike omain Ω⊂ℝn with values in the Clifford lgebra C𝓁0,n which are polymonogenic with respect to the (left) Dirac operator D=∑j=1n ej∂/∂xj, i.e. they belong to the kernel of Dk.We prove that any polymonogenic function f has a ecomposition of the form f=f1+xf2+···+xk−1fk , where x=x1e1+···+xnen 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∣−αxD, α∈ℝ\{0}. Copyright © John Wiley & Sons, Ltd.
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