New properties of harmonic polynomials
type: Item
instance of: scholarly article
title: New properties of harmonic polynomials
zbMATH Open document ID: 0811.41025
DOI: 10.1007/BF01262184
published in: Mathematical Notes
publication date: 1994-04-12T00:00:00Z
review text: Let \(H^ t_ n\) be the linear space of forms of degree \(t\); \(\text{Harm}^ t_{m,n}\) be the linear subspace in \(H^ t_ n\) consisting of those homogeneous polyharmonic polynomials \(\phi\) of order \(m\) and degree \(t\) that satisfy the polyharmonic equation \(D_{\sigma^ m}(\phi)= 0\). The main result (theorem 2) gives a dual description of polyharmonic polynomials, namely, \(\phi\in H^ t_ n\) is equivalent to the assertions \(\phi\in \text{Harm}^ t_{m,n}\) and \(D_ \phi(\sigma^ p)= 0\) for the same range of \(p\). To prove this result it is established (theorem 3) for \(\phi\in \text{Harm}^ t_{m,n}\) (harmonic polynomials) the representation of \(D_ \phi(\sigma^{t+p})\) via \(\sigma^ p\phi\), which is of independent interest. An application of the main result is also given (theorem 4).
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 525677
author: Georgij G. Rasputin
reviewed by: M.-B. A. Babaev
MaRDI profile type: Publication
cites work: Q4124934
cites work: Q3323132
description: scientific article; zbMATH DE number 525677