On the completeness of the exponential system in nonconvex domains

I. S. Galimov

Sbornik. Mathematics, Tome 27 (1975) no. 1, pp. 39-50

Résumé

Let $L(\lambda)=\sum_{j=1}^r A_je^{\lambda a_j}$, where $a_j$ ($1\leqslant j \leqslant r$) are the vertices of a convex polygon $\overline D$, and let $\{\lambda_\nu\}_{\nu=1}^\infty$ be the sequence of all of the zeros (which we assume to be simple) of $L(\lambda)$. Define $\Gamma\stackrel{\mathrm{df}}=\bigcup_{j=1}^r[0,a_j]$. For the system $\{e^{\lambda_\nu z}\}_{\nu=1}^\infty$, we construct a system of functions $\{\psi_\nu^*(z)\}_{\nu=1}^\infty$ which has the biorthogonality property on $\Gamma$. With the aid of the system $\{\psi_\nu^*(z)\}_{\nu=1}^\infty$, we construct the Dirichlet series for a function $f(z)$ which is continuous on $\Gamma$. We prove the following uniqueness theorem: If all the coefficients of the series are zero, then $f(z)\equiv0$. It follows from this theorem that the system $\{\psi_\nu^*(z)\}_{\nu=1}^\infty$ is complete outside of $\Gamma$. Bibliography: 3 titles.

Bibliographie

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[3] A. F. Leontev, “O predstavlenii funktsii posledovatelnostyami polinomov Dirikhle”, Matem. sb., 70 (112) (1966), 132–144 | MR