On Extremal Functions in Inequalities for Entire Functions

Let \(B_{\sigma}\) , \(\sigma>0\) , be the class of entire functions of exponential type \(\le\sigma\) bounded on the real line. For a number \(\tau\in\mathbb{R}\) and a sequence \(\{c_k\}_{k\in\mathbb{Z}}\) of complex numbers satisfying the condition \(\sum_{k\in\mathbb{Z}}|c_k|<+\infty\) , the operator \(H\) on \(B_{\sigma}\) defined by \(H(f)(x)=\sum_{k\in\mathbb{Z}}c_k f\biggl(x-\tau+\frac{k\pi}{\sigma}\biggr)\) is considered. Obviously, \(|H(f)(x)|\le \varkappa \|f\|_{\infty}, \qquad x\in\mathbb{R}, \quad f\in B_{\sigma}, \quad \varkappa=\sum_{k\in\mathbb{Z}} |c_k|.\) The main purpose of the paper is to describe all extremal functions for this inequality. Theorem 1 proved in the paper asserts that if (1) \(\overline{c_{s}}c_{s+1}<0\) for some \(s\in\mathbb{Z}\) and (2) there exists an \(\varepsilon\in\mathbb{C}\) with \(|\varepsilon|=1\) such that \(\varepsilon c_k (-1)^k\ge0\) for all \(k\in\mathbb{Z}\) , then the set of all extremal functions for the above inequality coincides with the set of functions of the form \(f(t)=\mu e^{i\sigma t}+\nu e^{-i\sigma t}\) , \(\mu,\nu\in\mathbb{C}\) . The proof of Theorem 1 essentially uses Theorem 2, which says that if \(f\in B_{\sigma}\) and there exists a point \(\xi\in\mathbb{R}\) for which \(|f(\xi)|=\|f\|_{\infty}\) and \(f(\xi+\pi/\sigma)=-f(\xi)\) , then \(f(t)=\mu e^{i\sigma t}+\nu e^{-i\sigma t}\) , \(\mu,\nu\in\mathbb{C}\) . Theorem 3 gives general examples of operators satisfying both conditions of Theorem 1. In particular, such is the fractional derivative operator \(H(f)(x)=f^{(r,\beta)}(x)\) for \(r\ge1\) and \(\beta\in\mathbb{R}\) .