Extremal Problems for Positive-Definite Bandlimited Functions. II. Eventually Negative Functions
Extremal Problems for Positive-Definite Bandlimited Functions. II. Eventually Negative Functions
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正定带限函数的极值问题。
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发表时间：
1983
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通讯作者：
B. Logan
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作者：
B. Logan
如果一个函数是指数型$leq lambda$的整个函数的实数行的限制，则该函数的带宽限制为$[-lambda，lambda]$。这类函数包括傅立叶变换在$[-lambda，lambda]$之外消失的所有函数。实值函数是正定的，如果它的傅里叶变换在实线上是非负的。这样的函数必然是偶数的。本文考虑连实值函数$f(T)$带限于$[-1，1]$。这些是形式为[f(T)=int_0^1{cos xt，df(X)，}]的函数，其中$df(X)$是有界Stieltjes测度。我们假设$f(0)=int_0^1{df(X)=1}$。我们证明了如果$df(X)geq 0$对$0 leq x leq varepsilon$对某个$varepsilon>；0$成立，则$f(T)$可以满足[f(T)leq 0 quad{ext{for}}|t|geq T]当且仅当$T geq pi$；对于$T=pi$当且仅当$f(T)$是正定函数[f(T)=frac{{(cos{t/2})^2}}{1-t^2}/{pi^2}=frac{pi}{2}int_0^1...
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 0quad { ext{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 ...