Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Weinstein transform; Hardy's type theorem; Cowling-Price's theorem; Beurling's theorem; Miyachi's theorem; Donoho-Stark's uncertainty principle
Weinstein transform; Hardy's type theorem; Cowling-Price's theorem; Beurling's theorem; Miyachi's theorem; Donoho-Stark's uncertainty principle
Summary:
The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi's theorem, Beurling's theorem, and Donoho-Stark's uncertainty principle are obtained for the Weinstein transform.
References:
[1] Nahia, Z. Ben, Salem, N. Ben: Spherical harmonics and applications associated with the Weinstein operator. Potential Theory---Proc. ICPT 94 J. Král et al. (1996), 235-241. MR 1404710
[2] Nahia, Z. Ben, Salem, N. Ben: On a mean value property associated with the Weinstein operator. Potential Theory---Proc. ICPT 94 J. Král et al. (1996), 243-253. MR 1404711
[3] Benedicks, M.: On Fourier transforms of functions supported on sets of finite Lebesgue measure. J. Math. Anal. Appl. 106 (1985), 180-183. DOI 10.1016/0022-247X(85)90140-4 | MR 0780328 | Zbl 0576.42016
[4] Beurling, A.: The Collected Works of Arne Beurling, Vol. 1, Vol. 2. L. Carleson, P. Malliavin, J. Neuberger, J. Wermen Birkhäuser Boston (1989). MR 1057613
[5] Brelot, M.: Equation de Weinstein et potentiels de Marcel Riesz. Lect. Notes Math. 681. Séminaire de Théorie de Potentiel Paris, No. 3 (1978), 18-38 French. MR 0521776 | Zbl 0386.31007
[6] Bonami, A., Demange, B., Jaming, P.: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoam. 19 (2003), 22-35. MR 1993414
[7] Cowling, M. G., Price, J. F.: Generalisations of Heisenberg's Inequality. Lect. Notes Math., Vol. 992 Springer Berlin (1983), 443-449. DOI 10.1007/BFb0069174 | MR 0729369 | Zbl 0516.43002
[8] Daher, R., Kawazoe, T., Mejjaoli, H.: A generalization of Miyachi's theorem. J. Math. Soc. Japan 61 (2009), 551-558. DOI 10.2969/jmsj/06120551 | MR 2532900 | Zbl 1235.43008
[9] Donoho, D. L., Stark, P. B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49 (1989), 906-931. DOI 10.1137/0149053 | MR 0997928 | Zbl 0689.42001
[10] Hardy, G. H.: A theorem concerning Fourier transform. J. Lond. Math. Soc. 8 (1933), 227-231. DOI 10.1112/jlms/s1-8.3.227 | MR 1574130
[11] Hörmander, L.: A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Mat. 29 (1991), 237-240. DOI 10.1007/BF02384339 | MR 1150375
[12] Mejjaoli, H., Trimèche, K.: An analogue of Hardy's theorem and its $L^p$-version for the Dunkl-Bessel transform. J. Concr. Appl. Math. 2 (2004), 397-417. MR 2224912
[13] Mejjaoli, H.: An analogue of Beurling-Hörmander's theorem for the Dunkl-Bessel transform. Fract. Calc. Appl. Anal. 9 (2006), 247-264. MR 2305441 | Zbl 1210.44002
[14] Mejjaoli, H., Trimèche, K.: A variant of Cowling-Price's theorem for the Dunkl transform on $\mathbb R$. J. Math. Anal. Appl. 345 (2008), 593-606. DOI 10.1016/j.jmaa.2006.02.029 | MR 2429159
[15] Mejjaoli, H.: Global well-posedness and scattering for a class of nonlinear Dunkl-Schrödinger equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 1121-1139. DOI 10.1016/j.na.2009.08.045 | MR 2577514 | Zbl 1185.35262
[16] Mejjaoli, H.: Nonlinear Dunkl-wave equations. Appl. Anal. 89 (2010), 1645-1668. DOI 10.1080/00036811.2010.495334 | MR 2773341 | Zbl 1204.35121
[17] Miyachi, A.: A generalization of theorem of Hardy. Harmonic Analysis Seminar, Izunagaoka, Shizuoka-Ken, Japan 1997 44-51 ().
[18] Morgan, G. W.: A note on Fourier transforms. J. Lond. Math. Soc. 9 (1934), 188-192. MR 1574180 | Zbl 0009.24801
[19] Parui, S., Sarkar, R. P.: Beurling's theorem and $L^{p}$-$L^{q}$ Morgan's theorem for step two nilpotent Lie groups. Publ. Res. Inst. Math. Sci. 44 (2008), 1027-1056. DOI 10.2977/prims/1231263778 | MR 2477903
[20] Pfannschmidt, C.: A generalization of the theorem of Hardy: A most general version of the uncertainty principle for Fourier integrals. Math. Nachr. 182 (1996), 317-327. DOI 10.1002/mana.19961820114 | MR 1419899 | Zbl 0873.42005
[21] Ray, S. K., Sarkar, R. P.: Cowling-Price theorem and characterization of heat kernel on symmetric spaces. Proc. Indian Acad. Sci. (Math. Sci.) 114 (2004), 159-180. DOI 10.1007/BF02829851 | MR 2062397 | Zbl 1059.22010
[22] Slepian, D., Pollak, H. O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty I. Bell. System Tech. J. 40 (1961), 43-63. DOI 10.1002/j.1538-7305.1961.tb03976.x | MR 0140732 | Zbl 0184.08601
[23] Slepian, D.: Prolate spheroidal wave functions, Fourier analysis and uncertainty IV: Extensions to many dimensions, generalized prolate spheroidal functions. Bell. System Tech. J. 43 (1964), 3009-3057. DOI 10.1002/j.1538-7305.1964.tb01037.x | MR 0181766 | Zbl 0184.08604
Search
Partner of
