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A representation of symmetric functions in Carleman-Gevrey spaces

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Abstract

For a symmetric function t(x)(x∈ℝd) one investigates the representation

, where δj(x) is the elementary symmetric polynomial of degree j. Let\(\bar \Omega \) be the closure of the domain Ω in ℝd, let be a numerical sequence such that ϕ(n) does not decrease, let be the Carleman-Gevrey space, i.e. the collection of functions ϕ(n+1)/ϕ(n) such that for any bounded subdomain\(K^\varphi \left( {\bar \Omega } \right)\) there exists a constant t∈C∞(Ω) Ω′⊂Ω with which one has the inequality ∣∂ ∝x t(x)∣⩽H∣∝∣+1∣∝∣!ϕ(∣∝∣) (∀x∈*#x03A9;'∀∝). Let S be the image of ℝd under the mapping x→(δ1(x), ..., δd(x)). One proves the following theorem: For any t∈kϕ(ℝd) there exists

such that

, if and only if ψ(n)⩾ϕ(nd)εn+1, where ε is some positive number, independent of n.

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Authors
  1. M. D. Bronshtein
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 116–126, 1986.

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Bronshtein, M.D. A representation of symmetric functions in Carleman-Gevrey spaces. J Math Sci 42, 1621–1628 (1988). https://doi.org/10.1007/BF01665049

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