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Generalized exponential series for distributions. (English. Russian original) Zbl 0893.30003
Dokl. Math. 53, No. 2, 266-269 (1996); translation from Dokl. Akad. Nauk 347, No. 6, 739-742 (1996).
Absolutely representing systems of generalized exponentials are considered which admit nontrivial expansions of zero in the space \(D_{\beta}'([-a,a]^p)\) of distributions on the \(p\)-dimensional cube \([-a,a]^p .\) The author finds conditions under which the existence of a nontrivial expansion of zero in a system of generalized exponentials \(\mathcal E_{\Lambda} \) that is absolutely convergent in \(D_{\beta}'([-a,a]^p)\) imply that \(\mathcal E_{\Lambda} \) is an absolutely representing system in \(D_{\beta}'([-a,a]^p).\) He also deals with the problem of existence of a continuous linear right inverse of the representation operator associated to an absolutely representing system \(\mathcal E_{\Lambda} \) of generalized exponentials.
Reviewer: F.Haslinger (Wien)
30B50 Dirichlet series, exponential series and other series in one complex variable
46A35 Summability and bases in topological vector spaces