Abstract
We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair h_1, h_2 of zonal harmonic functions such that f=h_1-bar{x} h_2. We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.
Highlights
In this note we prove a quaternionic Almansi-type Theorem for polynomials with quaternionic coefficients and, more generally, for slice-regular functions of a quaternionic variable
We show that to every slice-regular function f defined on a domain Ω of the skew field H of quaternions, it is possible to associate two uniquely defined harmonic functions h1, h2 such that f (x) = h1(x)−xh2(x) on Ω
An interesting aspect of these last results is their validity over any sphere in Ω, without requiring any symmetry with respect to the real axis
Summary
In this note we prove a quaternionic Almansi-type Theorem for polynomials with quaternionic coefficients and, more generally, for slice-regular functions of a quaternionic variable (see Sect. 3 for definitions and references about this function theory). In this note we prove a quaternionic Almansi-type Theorem for polynomials with quaternionic coefficients and, more generally, for slice-regular functions of a quaternionic variable We show that to every slice-regular function f defined on a domain Ω of the skew field H of quaternions, it is possible to associate two uniquely defined harmonic functions h1, h2 such that f (x) = h1(x)−xh2(x) on Ω. This result allows to apply well-known results of harmonic function theory on R4 to obtain a mean value property for slice-regular functions and a Poisson-type formula over three-dimensional spheres. This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen. ∗Corresponding author
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