论文标题
在傅立叶变换的（DIS）连续性上
On the (dis)continuity of the Fourier transform of measures
论文作者
论文摘要
在本文中，我们将研究有关模糊拓扑的傅立叶变换的连续性。我们表明，傅立叶变换在R上隐约不连续，但是在限制一类傅立叶变换度量时会变得连续，以使措施或它们的傅立叶变换是等电译本的。我们讨论产品和规范拓扑中傅立叶变换的连续性。我们表明，积极的确定度量的模糊收敛性意味着傅立叶变换的等电流量有界性，这解释了正面确定度量的圆锥体上傅立叶变换的连续性。在附录中，我们表征了一个任意LCAG中集合a措施的模糊预交性，以及一个组以定义自相关度量的第二可数字属性的必要性。
In this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of the Fourier transform on the cone of positive definite measures. In the appendix, we characterize vague precompactness of a set a measures in arbitrary LCAG, and the necessity of second countability property of a group for defining the autocorrelation measure.
