Bases of exponential functions in the spaces~$E^p$ on convex polygons

A. M. Sedletskii

Izvestiya. Mathematics , Tome 13 (1979) no. 2, pp. 387-404

Résumé

Let $D$ be a convex polygon in the complex plane; let $a_1,a_2,\dots,a_m$ $(m\geq 3)$ be its vertices, numbered in the order of a circuit around $D$ in the positive direction; let $\varphi_k=\arg(a_{k+1}-a_k)-\pi/2$; and let $2l_k$ be the length of the edge $a_k$, $a_{k+1}$. Let $\Lambda=\Lambda_1\cup\Lambda_2\cup\dots\cup\Lambda_m$, where $$ \Lambda_k=\biggl\{l^{-1}_ke^{-i\varphi_k}\biggl(\pi n+\frac\pi2+\alpha_k+\varepsilon_{kn}\biggr)\biggr\}_{n=0}^{+\infty},\quad k=1,2,\dots,m. $$ If $\alpha_1+\dots+\alpha_m=0$ and $\{\varepsilon_{kn}\}\in l^2$ for $p\geqslant2$ and $\{\varepsilon_{kn}\}\in l^p$ for $1$, $ k=1,2,\dots,m$, then $\{\exp(\lambda_nz)\}$, $\lambda_n\in\Lambda$, is a basis in the space $E^p(D)$, $1$. Bibliography: 16 titles.