Extremal Problems for Positive-Definite Bandlimited Functions. II. Eventually Negative Functions

Abstract

A function is bandlimited to $[ - \lambda ,\lambda ]$ if it is the restriction to the real line of an entire function of exponential type $ \leq \lambda $. This class of functions includes all functions whose Fourier transforms vanish outside $[ - \lambda ,\lambda ]$. A real-valued function is positive definite if its Fourier transform is nonnegative on the real line. Such a function is necessarily even. In this paper we consider even real-valued functions $f(t)$ bandlimited to $[ - 1,1]$. These are functions of the form \[ f(t) = \int_0^1 {\cos xt\,dF(x),} \]where $dF(x)$ is a bounded Stieltjes measure. We suppose that $f(0) = \int_0^1 {dF(x) = 1} $. We show that if $dF(x) \geq 0$ for $0 \leq x \leq \varepsilon $ for some $\varepsilon > 0$, then $f(t)$ can satisfy \[ f(t) \leq 0\quad {\text{for }}| t | \geq T \] if and only if $T \geq \pi $; and for $T = \pi $ if and only if $f(t)$ is the positive-definite function \[ f(t) = \frac{{(\cos {t / 2})^2 }}{{{{1 - t^2 } / {\pi ^2 }}}} = \frac{\pi }{2}\int_0^1 ...