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La surjectivité de l’application moyenne pour les espaces préhomogènes. (English) Zbl 0557.43007
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.
Reviewer: J.Faraut

MSC:
 43A85 Harmonic analysis on homogeneous spaces 46F10 Operations with distributions and generalized functions
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References:
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