Abstract: | For a symmetric function t(x)(x![isin](/content/u30871327658qp47/xxlarge8712.gif) d) one investigates the representation, where j(x) is the elementary symmetric polynomial of degree j. Let
be the closure of the domain in d, let be a numerical sequence such that (n) does not decrease, let be the Carleman-Gevrey space, i.e. the collection of functions (n+1)/ (n) such that for any bounded subdomain
there exists a constant t C ( ) ![OHgr](/content/u30871327658qp47/xxlarge937.gif) ![prime](/content/u30871327658qp47/xxlarge8242.gif) ![sub](/content/u30871327658qp47/xxlarge8834.gif) with which one has the inequality ![mid](/content/u30871327658qp47/xxlarge8739.gif)
x
t(x)![mid](/content/u30871327658qp47/xxlarge8739.gif) H![mid](/content/u30871327658qp47/xxlarge8739.gif) ![prop](/content/u30871327658qp47/xxlarge8733.gif) +1![mid](/content/u30871327658qp47/xxlarge8739.gif) ![prop](/content/u30871327658qp47/xxlarge8733.gif) ! (![mid](/content/u30871327658qp47/xxlarge8739.gif) ![prop](/content/u30871327658qp47/xxlarge8733.gif) ) ( x *#x03A9;'![forall](/content/u30871327658qp47/xxlarge8704.gif) ). Let S be the image of d under the mapping x ( 1(x), ..., d(x)). One proves the following theorem: For any t k ( d) there exists such that, if and only if (n)![ges](/content/u30871327658qp47/xxlarge10878.gif) (nd) n+1, where is some positive number, independent of n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 116–126, 1986. |