La surjectivité de l'application moyenne pour les espaces préhomogènes
type: Item
instance of: scholarly article
title: La surjectivité de l'application moyenne pour les espaces préhomogènes
zbMATH Open document ID: 0557.43007
DOI: 10.1016/0022-1236(85)90059-X
author: Hubert Rubenthaler
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: Let f be a homogeneous polynomial on \({\mathbb{R}}^ n\). For C a connected component of \(\{f(x)\neq 0\}\) and s a complex number one considers the integral \(Z(\phi,s)=\int_{C}\phi (x)| f(x)|^ sdx,\) where \(\phi\) is a test function. This integral converges for Re s\(>0\) and admits a meromorphic continuation. The author studies the distributions occurring as coefficients in the Laurent developments of the function \(s\mapsto Z(\phi,s)\) at its poles. He proves a generalization of a theorem of Borel, which corresponds to the special case \(f(x)=x\), \(n=1\), and says: for any sequence \(c_ k\) of complex numbers there exists a \(C^{\infty}\) function \(\phi\) on \({\mathbb{R}}\) such that \(\phi^{(k)}(0)=c_ k.\) Let \(M_{\phi}(t)\) be the integral of \(\phi\) on \(\{f(x)=t\}\cap C.\) As a corollary of the previous result, since Z(\(\phi\),s) is the Mellin transform of \(M_{\phi}(t)\), the space of the functions \(M_{\phi}\) can be described in terms of asymptotic developments at 0.
reviewed by: Jacques Faraut
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46F10
zbMATH DE Number: 3888306
zbMATH Keywords: mean map
zbMATH Keywords: homogeneous polynomial
zbMATH Keywords: theorem of Borel
zbMATH Keywords: asymptotic developments
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3888306