We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with poly-harmonic functions, the elements of the kernel of an iterated Laplacian. Here we consider polynomials of the form $P(x)=sum_{k=0}^d x^ka_k$, with Clifford coefficients $a_kinmathbb R_{n}$, and get an analogous decomposition related to zonal polyharmonics. We show the relation between such decomposition and the Dirac (or Cauchy-Riemann) operator and extend the results to slice-regular functions.

Almansi-type theorems for slice-regular functions on Clifford algebras / Perotti, Alessandro. - In: COMPLEX VARIABLES AND ELLIPTIC EQUATIONS. - ISSN 1747-6933. - 2020/66:8(2020), pp. 1287-1297. [10.1080/17476933.2020.1755967]

Almansi-type theorems for slice-regular functions on Clifford algebras

Perotti, Alessandro
2020

Abstract

We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with poly-harmonic functions, the elements of the kernel of an iterated Laplacian. Here we consider polynomials of the form $P(x)=sum_{k=0}^d x^ka_k$, with Clifford coefficients $a_kinmathbb R_{n}$, and get an analogous decomposition related to zonal polyharmonics. We show the relation between such decomposition and the Dirac (or Cauchy-Riemann) operator and extend the results to slice-regular functions.
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Perotti, Alessandro
Almansi-type theorems for slice-regular functions on Clifford algebras / Perotti, Alessandro. - In: COMPLEX VARIABLES AND ELLIPTIC EQUATIONS. - ISSN 1747-6933. - 2020/66:8(2020), pp. 1287-1297. [10.1080/17476933.2020.1755967]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/257858
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