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A qualitative uncertainty principle for hypergroups. (English) Zbl 0781.43003

Functional analysis and operator theory, Proc. Conf. Mem. U. N. Singh, New Delhi/India 1990, Lect. Notes Math. 1511, 1-9 (1992).
[For the entire collection see Zbl 0745.00065.]
Uncertainty principles in Fourier analysis assert that the more a function is concentrated, the more its Fourier transform is spread out. Much work has been done in generalizing such principles to locally compact groups, in particular abelian ones, and, recently, to Gelfand pairs and commutative hypergroups. Let \(K\) be a commutative hypergroup and \(\widehat{K}\) its dual space, that is, the set of all bounded continuous characters of \(K\). \(K\) possesses a non-trivial Haar measure \(m\), and associated to \(m\) is a measure \(\pi\) on \(\widehat{K}\), the Plancherel measure, such that the Fourier transformation is an isometric isomorphism between \(L^ 2(K,m)\) and \(L^ 2(\widehat{K},\pi)\). For \(f\in L^ 1(K,m)\), let \(A_ f = \{x\in K: f(x) \neq 0\}\) and \(B_ f = \{\gamma\in\widehat{K}:\widehat{f}(\gamma) \neq 0\}\). Generalizing well- known uncertainty principles for abelian groups to the hypergroup setting, the author proves for \(f \in L^ 1(K,m)\): (i) \(m(A_ f)\pi(B_ f)<1\) implies \(f=0\) a.e. (ii) If the connected component of the identity of \(K\) is non-compact, and if a certain continuity condition for the measure of translates of sets in \(K\) is satisfied, then \(f = 0\) a.e. provided that \(m(A_ f) < \infty\) and \(\pi(B_ f) < \infty\).
For locally compact abelian groups \(G\) there is a much stronger result then (i) due to K. Smith [SIAM J. Appl. Math. 50, 875-882 (1990; Zbl 0699.43006)] as follows. Let \(\varepsilon,\delta\geq 0\), and let \(T\) and \(U\) be Borel sets in \(G\) and \(\widehat{G}\), respectively. If \(f\in L^ 1(G)\) is \(\varepsilon\)-concentrated on \(T\) and \(\widehat{f}\) is \(\delta\)-concentrated on \(U\), then \(m(T)\pi(U) \geq (1-\varepsilon- \delta)/(1+\delta)\). This latter result has very recently been extended to commutative hypergroups by M. Voit [Math. Nachr. 164, 187-194 (1993)]. If \(\varepsilon = \delta = 0\), this reduces to (i). The technical continuity condition required in (ii) is difficult to verify and, as pointed out by Voit [loc. cit.] fails to hold for certain classes of commutative hypergroups.

MSC:

43A62 Harmonic analysis on hypergroups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A80 Analysis on other specific Lie groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A05 Measures on groups and semigroups, etc.
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