基于Clifford-Fourier变换的不确定原理
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摘要
在Clifford分析框架下,考虑一种基于超李代数■的Clifford-Fourier变换,该超李代数包含经典李代数sl_2为其偶子代数。介绍定义以及已有的相关性质,研究该变换与经典Fourier变换类似的性质,如微分公式、乘法公式、Plancherel定理以及Parsevel等式等。根据Holder不等式以及前面推导的结论,证明了Heisenberg-Pauli-Weyl型不确定原理。
In the framework of Clifford analysis, a generalized Clifford-Fourier transform is considered. This transform is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra ■(containing Lie algebra sl_2 as its even subalgebra). Some further properties of the Clifford-Fourier transform, such as differential formula, multiplication formula, Plancherel Theorem and Parsevel's Identity, are developed. As an application, a Heisenberg-Pauli-Weyl type uncertainty principle for the Clifford-Fourier transform is proven.
In the framework of Clifford analysis, a generalized Clifford-Fourier transform is considered. This transform is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra ■(containing Lie algebra sl_2 as its even subalgebra). Some further properties of the Clifford-Fourier transform, such as differential formula, multiplication formula, Plancherel Theorem and Parsevel's Identity, are developed. As an application, a Heisenberg-Pauli-Weyl type uncertainty principle for the Clifford-Fourier transform is proven.
引文
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[6] KAWAZOE T,MEJJAOLI H.Uncertainty principles for the Dunkl transform [J] .Hiroshima Mathematical Journal,2010,20(4):241-268.
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[14] BRACKX F,DELANGHE R,SOMMEN F.Clifford analysis [M].Boston,MA:Pitman (Advanced Publishing Program),1982.
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[19] KOU K I,QIAN T.The Paley-Wiener theorem in Rn with the Clifford analysis setting [J].Journal of Functional Analysis,2002,189(1):227-241.
[20] BRACKX F,DE SCHEPPER N,SOMMEN F.The Clifford-Fourier transform [J].Journal of Fourier Analysis and Applications,2005,11(6):669-681.
[21] DE BIE H,XU Y.On the Clifford-Fourier transform [J].International Mathematics Research Notices,2011,22:5123-5163.
[22] WATSON G N.A treatise on the theory of bessel functions [M].Cambridge:Cambridge University Press,1944.
[23] YANG Y,QIAN T.Schwarz lemma in Euclidean spaces [J].Complex Variables and Elliptic Equations,2006,51(7):653-659.
[2] FOLLAND G B,SITARAM A.The uncertainty principle:a mathematical survey [J].Journal of Fourier Analysis and Applications,1997,26(13):207-138.
[3] BAHRI M,HITZER E.Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra Cl3,0[J].Advances in Applied Clifford Algebras,2006,16(1):41-61.
[4] DE JEU M F E.An uncertainty principle for integral operators [J] .Journal of Functional Analysis,1994,122(1):247-253.
[5] FU Y X,LI L Q.Uncertainty principle for multivector-valued functions [J] .International Journal of Wavelets,Multiresolution and Information Processing,2015,13(1):1550005.
[6] KAWAZOE T,MEJJAOLI H.Uncertainty principles for the Dunkl transform [J] .Hiroshima Mathematical Journal,2010,20(4):241-268.
[7] R?SIER M.An uncertainty principle for the Dunkl transform [J].Bulletin of the Australian Mathematical Society,1999,59(3):353-360.
[8] R?SIER M,VOIT M.An uncertainty principle for Hankel transforms [C].Proceedings of the American Mathematical Society,1999,127(1):183-194.
[9] SELIG K K.Uncertainty principles revisited [J].Electronic Transactions on Numerical Analysis,2002,14:165-177.
[10] ABREU L D.Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions [J].Applied and Computational Harmonic Analysis,2010,29(3):287-302.
[11] BALAN R.Density and redundancy of the noncoherent Weyl-Heisenberg superframes [C].The Functional and Harmonic Analysis of Wavelets and Frames.San Antonio,1999,247:29-41.
[12] BALAN R.Multiplexing of signals using superframes [C].SPIE Wavelets Applications of Signal and Image Processing.2000,4119:118-129.
[13] GR?CHENIG K,LYUBARSKII Y.Gabor (super)frames with Hermite functions [J].Mathematische Annalen,2009,345(2):267-286.
[14] BRACKX F,DELANGHE R,SOMMEN F.Clifford analysis [M].Boston,MA:Pitman (Advanced Publishing Program),1982.
[15] DELANGHE R,SOMMEN F,SOUCEK V.Clifford algebra and spinor valued functions,a function theory for Dirac operator,Kluwer [M].Dordrecht,1992.
[16] HOWE R ,TAN E C.Nonabelian harmonic analysis [M].New York:Universitext,Springer,1992.
[17] BRACKX F,DE SCHEPPER N,SOMMEN F.The two-dimensional Clifford-Fourier transform [J].Journal of Mathematical Imaging and Vision,2006,26(1-2):5-18.
[18] BRACKX F,DE SCHEPPER N,SOMMEN F.The Fourier transform in Clifford analysis [C].Advances in Imaging and Electron Physics,2009.
[19] KOU K I,QIAN T.The Paley-Wiener theorem in Rn with the Clifford analysis setting [J].Journal of Functional Analysis,2002,189(1):227-241.
[20] BRACKX F,DE SCHEPPER N,SOMMEN F.The Clifford-Fourier transform [J].Journal of Fourier Analysis and Applications,2005,11(6):669-681.
[21] DE BIE H,XU Y.On the Clifford-Fourier transform [J].International Mathematics Research Notices,2011,22:5123-5163.
[22] WATSON G N.A treatise on the theory of bessel functions [M].Cambridge:Cambridge University Press,1944.
[23] YANG Y,QIAN T.Schwarz lemma in Euclidean spaces [J].Complex Variables and Elliptic Equations,2006,51(7):653-659.