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On the uncertainty principle in harmonic analysis. (English) Zbl 1025.42004

Byrnes, James S. (ed.), Twentieth century harmonic analysis–a celebration. Proceedings of the NATO Advanced Study Institute, Il Ciocco, Italy, July 2-15, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 33, 3-29 (2001).
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 B. J√∂ricke [“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.
For the entire collection see [Zbl 0972.00019].

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions

Citations:

Zbl 0827.42001
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