Completeness of the exponential system on a segment of the real axis

A. M. Gaisin ; B. E. Kanguzhin ; A. A. Seitova

Eurasian mathematical journal, Tome 13 (2022) no. 2, pp. 37-42

Résumé

Let $\Lambda=\{\lambda_n\}$ be the sequence of all zeros of the entire function $\Delta(\lambda)=1-i\lambda\int_0^1f(t)e^{i\lambda t}dt$ of exponential type. We consider exponential system of functions $e(\Lambda)=\{t^{p-1}e^{i\lambda_nt}, 1\leqslant p\leqslant m_n\}$, where $m_n$ — is the multiplicity of the zero $\lambda_n$. The question is: for which $a$, $b$ ($a$) is the system $e(\Lambda)$ complete (incomplete) in the space $L^2(a, b)$? Let $D$ be the length of the indicator conjugate diagram of the entire function $\Delta(\lambda)$. Then the following statements are valid: when $b-a>D$ the system $e(\Lambda)$ is incomplete in $L^2(a,b)$; when $b-a$ the system $e(\Lambda)$ is complete in $L^2(a,b)$; if we remove from $\Lambda$ any two points $\lambda$ and $\mu$, then the system $e(\Omega)$, $\Omega=\Lambda\setminus\{\lambda,\mu\}$ is incomplete in $L^2(a,b)$ also when $b-a=D$.