Sur la synthèse spectrale dans les espaces symétriques. (On spectral synthesis in symmetric spaces) (Q1097478)

The principal theme of the article is the localization of ideals in algebras \(A_ q({\mathbb{C}}^ n)\) of entire functions f in \({\mathbb{C}}^ n\) satisfying the following growth restriction: \[ | f(\zeta)| \leq A \exp (Bq(\zeta)),\quad q(\zeta):=\sum_{1\leq k\leq n}| \zeta_ k|^{\alpha_ k},\quad \zeta =(\zeta_ 1,...,\zeta_ n)\in {\mathbb{C}}^ n,\quad \alpha_ k>0, \] A and B being positive constants depending on f. The algebra \(A_ q({\mathbb{C}}^ n)\) is endowed with a natural topology. It is shown that \(A_ q\) contains a finite family of functions without common zeros and such that 1 does not belong to the closure of the ideal they generate in \(A_ q\). This is an extension (and an application as well) of a well known result of \textit{D. I. Gurevich} [Funct. Anal. Appl. 9, 116-120 (1975; Zbl 0326.46020)]. There are also some results concerning non localizable ideals in algebras connected with symmetric spaces of non compact type and of rank\(\geq 2\).