关于Fourier变换的正性
On the positivity of Fourier transforms
摘要：
特征的建设性的方式设置的实函数的傅里叶变换是积极的似乎是一个开放的问题。一些充分条件是已知的，但它们远不是穷举的。我们提出了两个建设性的必要条件的傅立叶变换的积极性，并测试他们的能力，约束的积极性域。一个使用解析延拓和詹森不等式和其他处理Toeplitz决定因素和Bochner定理。应用进行了讨论，包括扩展到二维傅里叶-贝塞尔变换和问题的正互易性，即正的功能与正变换。
英文摘要：
Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive sets of necessary conditions for positivity of the Fourier transforms and test their ability of constraining the positivity domain. One uses analytic continuation and Jensen inequalities and the other deals with Toeplitz determinants and the Bochner theorem. Applications are discussed, including the extension to the two-dimensional Fourier-Bessel transform and the problem of positive reciprocity, i.e. positive functions with positive transforms.
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