On the uncertainty principle in harmonic analysis (Q2772766)

This expository paper addresses several issues of joint localization of a function and its Fourier transform. In some sense it represents a digest version of results considered in more detail in the joint monograph of the author with \textit{B. Jöricke} [``The uncertainty principle in harmonic analysis'' (1994; Zbl 0827.42001)]. The results are roughly categorized in terms of those requiring analytic function theory versus those that are real-variable in nature. They are decomposed further into a collection of the following seven localization properties (in the case of functions, or distributions, of a real variable): (1) (vanishing at infinity) \(f(t)=O(M(t))\) as \(|t|\to\infty\) where \(M(t)\to 0\) as \(|t|\to\infty\); (2) (one sided decay) \(f(t)=O(M(t))\), say, as \(t\to +\infty\); (3) deep zero: \(f(t)=O(M(t))\) as \(t\to t_0\) where \(M(t)\to 0\) as \(t\to t_0\); (4) (sparse support) \(f\) is supported on a set of small measure; (5) (gaps) \(f\) has large gaps in its support; (6) \(f\) has compact support or (7) \(f\) vanishes on a half line. These conditions on \(f\) have analogues, often with more intricate descriptions, in several variables. The central issue is the extent to which a condition from the collection (1)--(7) on \(f\) is compatible or incompatible with another condition from the set on its Fourier transform \(\hat f\). Several precise results concerning these issues are reviewed.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00019].