Geodesic
Representing systems of exponential functions in polycylindrical domains
Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 439-456 
Le Khaǐ Khoǐ; Yu. F. Korobeinik. Representing systems of exponential functions in polycylindrical domains. Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 439-456. http://geodesic.mathdoc.fr/item/SM_1985_50_2_a8/
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     author = {Le Khaǐ Khoǐ and Yu. F. Korobeinik},
     title = {Representing systems of exponential functions in polycylindrical domains},
     journal = {Sbornik. Mathematics},
     pages = {439--456},
     year = {1985},
     volume = {50},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1985_50_2_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The main result in this article is the following. Theorem. {\it Let $D_p$ $(1\leqslant p\leqslant m)$ be a bounded convex domain in the $z_p$-plane with support function $h_p(-\varphi),$ and let $\Lambda_p\overset{\mathrm{df}}=\{\lambda_k^{(p)}\}_{k=1}^\infty$ be zeros $($not necessarily simple$)$ of an exponential function $\mathscr L_p(\lambda)$ with indicator $h_p(\varphi)$ $($the function $\mathscr L_p(\lambda)$ may also have other zeros besides $\{\lambda_k^{(p)}\}_{k=1}^\infty,$ and$,$ moreover$,$ of arbitrary multiplicity$).$ Assume that $\mathscr E_{\Lambda_p}\overset{\mathrm{df}}=\{e^{\lambda_k^{(p)}z_p}\}_{k=1}^\infty$ is an absolutely representing system in $\mathscr H(D_p),$ $p=1,2,\dots,m$. Then $$ \mathscr E_{\Lambda}\overset{\mathrm{df}}=\big\{e^{\lambda_{k_1}^{(1)}z_1+\dots+\lambda_{k_m}^{(m)}z_m}\big\}_{k_1,\dots,k_m=1}^\infty $$ is an absolutely representing system in $\mathscr H(D),$ where $D=D_1\times D_2\times\dots\times D_m$ and $\mathscr H(G)$ is the space of holomorphic functions in a domain $G,$ with the topology of uniform convergence on compact subsets of $G$.} The properties of nontrivial expansions of zero in $\mathscr H(D)$ with respect to a system $\mathscr E_\Lambda$ are also studied. In particular, it is proved that if $D_p$, $\Lambda_p$, and $\mathscr L_p(\lambda)$ are the same as in the statement of the theorem, then $\mathscr E_\Lambda$ is an absolutely representing system in $\mathscr H(D)$ if and only if $\mathscr H(D)$ has a nontrivial expansion of zero with respect to the system $\mathscr E_\Lambda$. Bibliography: 9 titles.

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