Minimizing functions for an uncertainty principle on locally compact groups of bounded representation dimension

Kaniuth, Eberhard

doi:10.1090/S0002-9939-06-08451-6

Minimizing functions for an uncertainty principle on locally compact groups of bounded representation dimension
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Proc. Amer. Math. Soc. 135 (2007), 217-227 Request permission

Abstract:

Let $G$ be a locally compact group of bounded representation dimension $d(G)$. Then, for any integrable function $f$ on $G$, the product of the measures of the support of $f$ and the support of its operator-valued Fourier transform on the dual space of $G$ is bounded below by $1/d(G)$. We classify all functions for which equality holds and prove criteria for when such functions exist.

References

Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
David L. Donoho and Philip B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), no. 3, 906–931. MR 997928, DOI 10.1137/0149053
Siegfried Echterhoff and Eberhard Kaniuth, Certain group extensions and twisted covariance algebras with generalized continuous trace, Harmonic analysis (Luxembourg, 1987) Lecture Notes in Math., vol. 1359, Springer, Berlin, 1988, pp. 159–169. MR 974312, DOI 10.1007/BFb0086596
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
Wilfried Hauenschild and Eberhard Kaniuth, Harmonische Analyse auf Gruppen mit endlich-dimensionalen irreduziblen Darstellungen, Math. Z. 144 (1975), no. 3, 239–256 (German). MR 390125, DOI 10.1007/BF01214137
Jeffrey A. Hogan, A qualitative uncertainty principle for locally compact abelian groups, Miniconferences on harmonic analysis and operator algebras (Canberra, 1987) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 16, Austral. Nat. Univ., Canberra, 1988, pp. 133–142. MR 953989
Jeffrey A. Hogan, A qualitative uncertainty principle for unimodular groups of type $\textrm {I}$, Trans. Amer. Math. Soc. 340 (1993), no. 2, 587–594. MR 1102222, DOI 10.1090/S0002-9947-1993-1102222-4
Adam Kleppner and Ronald L. Lipsman, The Plancherel formula for group extensions. I, II, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459–516; ibid. (4) 6 (1973), 103–132. MR 342641
Gitta Kutyniok, A weak qualitative uncertainty principle for compact groups, Illinois J. Math. 47 (2003), no. 3, 709–724. MR 2007232
Gitta Kutyniok, A qualitative uncertainty principle for functions generating a Gabor frame on LCA groups, J. Math. Anal. Appl. 279 (2003), no. 2, 580–596. MR 1974047, DOI 10.1016/S0022-247X(03)00038-6
Tamás Matolcsi and József Szűcs, Intersection des mesures spectrales conjuguées, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A841–A843 (French). MR 326460
Calvin C. Moore, Groups with finite dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401–410. MR 302817, DOI 10.1090/S0002-9947-1972-0302817-8
D.J.S. Robinson, Finiteness conditions and solvable groups. I, Springer, Berlin-Heidelberg-New York, 1972.
Kennan T. Smith, The uncertainty principle on groups, SIAM J. Appl. Math. 50 (1990), no. 3, 876–882. MR 1050918, DOI 10.1137/0150051