Hermite interpolation and an inequality for entire functions of exponential type (Q1383256)

Let \(f\) be an entire function of exponential type \(<\pi\). A theorem of Carlson asserts that \(f\equiv 0\) if \(f(n)=0\) for every \(n\in\mathbb{Z}\). An extension of this result is obtained in this paper in terms of integral inequalities for entire functions \(f\) of exponential type. Theorem 1 Let \(m\in\mathbb{N}\), \(p>0\) and \(c\in[0,1)\), and \(\{\lambda_n\}\) be a sequence of real numbers such that \(|\lambda_n-n |\leq\Delta <\infty\), \(|\lambda_{n+u}-\lambda_n|\geq \delta >0\), \((u\neq 0)\). If \(\limsup_{y\to\infty} y^{-1} \log(| f(iy) |+ | f(-iy)|) =c\pi\), then there exists a constant \(B\) depending on \(m,p,c,\Delta\) and \(\delta\) such that \[ \int^\infty_{-\infty} \bigl| f(x) \bigr|^p dx<B\sum^\infty_{n=-\infty} \sum^{m-1}_{\mu=0} \bigl| f^{(\mu)} (\lambda_n) \bigr|^p. \] From Theorem 1 are derived a sampling theorem for irregular spaced points, and an extension of a theorem of R. J. Duffin and A. C. Schaeffer concerning entire functions of exponential type which are bounded at the points of the sequence \(\{\lambda_n\}\).