Completeness of the Exponential System on a Segment of the Real Axis

Abstract

Let \(\Lambda = \{\lambda_n\}\) be the sequence of all zeros of the entire function \(\Delta(\lambda) = 1 - i\lambda \int_0^1 f(t) e^{i \lambda t} dt\) of exponential type. We consider exponential system of functions \(e(\Lambda) = \{t^{p-1} e^{i \lambda_n t}, 1 \leq p \leq m_n\}\), where \(m_n\) is the multiplicity of the zero \(\lambda_n\). The question is: for which \(a, b\) (\(a < b\)) 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 \(b - a < D\) 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 \(b - a = D\).