Extremal Problems for Positive-Definite Bandlimited Functions. III. The Maximum Number of Zeros in an Interval $[0,T

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 positive definite functions bandlimited to $[ - 1,1]$. These are functions of the form \[ f(t) = \int_0^1 {\cos xt\,dF(x),\quad dF(x) \geq 0,}\] with $dF(x)$ a bounded Stieltjes measure. We suppose that $f(0) = 1$. Let $N(T)$ denote the number of zeros, counted according to multiplicity, in the closed interval $[0,T]$ of such a function $f(t)$. In this paper we show that \[ N(T) \leq \left[\kern-0.15em\left[ {\frac{2}{\pi }T} \right]\kern-0.15em\right]\] where $[\kern-0.15em[ x ]\kern-0.15em]$ denotes the largest integer contained in x, with equality attaining for $T = \frac{{n\pi }}{2}$ if, and only if, \[ f(t) = \left( {\cos \frac{t}{n}} \right)^n ;\] and equality may attain in countless ways for ${{n\pi } /{2 < T < (n + 1)}}{\pi / 2}$.