A note on Hardy’s theorem and rotations. (English) Zbl 1470.42014
Summary: We give a very short proof, using the Hermite semigroup, to a generalized version of Hardy’s theorem due to J. A. Hogan and J. D. Lakey [Proc. Am. Math. Soc. 134, No. 5, 1459–1466 (2006; Zbl 1089.42002)]. We characterize \(f\in L^2({\mathbb{R}}^n)\) when decay of \(f\) and its Fourier transform \({\hat{f}}\) is assumed in some rays of the complex plane. Also considering the decay of the Hermite coefficient of \(f\in L^2({\mathbb{R}}^n)\), we prove a version of Hardy’s theorem related to rotation.
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
Citations:
Zbl 1089.42002References:
| [1] | Cowling, M., Price, J.F.: Generalisations of Heisenberg’s inequality. In: Harmonic Analysis (Cortona, 1982), Lecture Notes in Math., vol. 992, pp. 443-449. Springer, Berlin (1983) · Zbl 0516.43002 |
| [2] | Folland, GB; Sitaram, A., The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3, 3, 207-238 (1997) · Zbl 0885.42006 · doi:10.1007/BF02649110 |
| [3] | Hardy, GH, A theorem concerning Fourier transforms, J. Lond. Math. Soc., 8, 3, 227-231 (1933) · Zbl 0007.30403 · doi:10.1112/jlms/s1-8.3.227 |
| [4] | Hogan, JA; Lakey, JD, Hardy’s theorem and rotations, Proc. Am. Math. Soc., 134, 5, 1459-1466 (2006) · Zbl 1089.42002 · doi:10.1090/S0002-9939-05-08098-6 |
| [5] | Hörmander, L., A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat., 29, 2, 237-240 (1991) · Zbl 0755.42009 · doi:10.1007/BF02384339 |
| [6] | Tang, S.; Niu, P., Hardy’s theorem and rotations of several complex variables, Anal. Math., 35, 4, 273-287 (2009) · Zbl 1212.32004 · doi:10.1007/s10476-009-0403-y |
| [7] | Thangavelu, S.: Hermite and Laguerre semigroups: some recent developments. In: Orthogonal Families and Semigroups in Analysis and Probability, Sémin. Congr., vol. 25, pp. 251-284. Soc. Math. France, Paris (2012) · Zbl 1295.22015 |
| [8] | Thangavelu, S., An Introduction to the Uncertainty Principle, Progress in Mathematics, vol. 217 (2004), Boston: Birkhäuser Boston Inc, Boston · Zbl 1188.43010 · doi:10.1007/978-0-8176-8164-7 |
| [9] | Vemuri, M.K.: Hermite expansions and Hardy’s theorem. arXiv preprint arXiv:0801.2234 (2008) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
