Fourier transform and related integral transforms in superspace. (English) Zbl 1149.30037
The author together with F. Sommen developed a theory of Clifford analysis in super space [e.g. Adv. Appl. Clifford Algebras 17, 357–382 (2007; Zbl 1129.30034)]. In super space the basic algebra is generated by \(m\) commuting real variables \(x_i\) and \(m\) orthogonal Clifford generators \(e_i\) as well as by \(2n\) anti-commuting variables \(x'_j\) and \(2n\) symplectic Clifford generators \(e'_j\). A super Dirac operator, a super Euler operator and a super Laplace operator may be defined within this setting, and a Clifford analysis can be developed.
Here the author studies a Fourier transform, a fractional Fourier transform, and a Radon transform in super space, where the Clifford theory gives rise to the definition of new kernels for these transforms. An eigenfunction basis for the Fourier Transform is constructed using special Hermite polynomials.
Here the author studies a Fourier transform, a fractional Fourier transform, and a Radon transform in super space, where the Clifford theory gives rise to the definition of new kernels for these transforms. An eigenfunction basis for the Fourier Transform is constructed using special Hermite polynomials.
Reviewer: Klaus Habetha (Aachen)
MSC:
| 30G35 | Functions of hypercomplex variables and generalized variables |
| 44A12 | Radon transform |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
Citations:
Zbl 1129.30034References:
| [1] | Berezin, F. A., The Method of Second Quantization, Pure Appl. Phys., vol. 24 (1966), Academic Press: Academic Press New York, translated from Russian by Nobumichi Mugibayashi and Alan Jeffrey · Zbl 0151.44001 |
| [2] | Berezin, F. A., Introduction to Algebra and Analysis with Anticommuting Variables (1983), Moskov. Gos. Univ.: Moskov. Gos. Univ. Moscow, with a preface by A.A. Kirillov · Zbl 0527.15020 |
| [3] | Brackx, F.; Delanghe, R.; Sommen, F., Clifford Analysis, Res. Notes Math., vol. 76 (1982), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, MA · Zbl 0529.30001 |
| [4] | H. De Bie, D. Eelbode, F. Sommen, On spherical harmonics in superspace and how they uniquely determine the Berezin integral, submitted for publication; H. De Bie, D. Eelbode, F. Sommen, On spherical harmonics in superspace and how they uniquely determine the Berezin integral, submitted for publication · Zbl 1179.30053 |
| [5] | H. De Bie, F. Sommen, A Cauchy integral formula in superspace, submitted for publication; H. De Bie, F. Sommen, A Cauchy integral formula in superspace, submitted for publication · Zbl 1179.30054 |
| [6] | De Bie, H.; Sommen, F., A Clifford analysis approach to superspace, Ann. Physics, 322, 2978-2993 (2007) · Zbl 1132.81017 |
| [7] | De Bie, H.; Sommen, F., Correct rules for Clifford calculus on superspace, Adv. Appl. Clifford Algebr., 17, 357-382 (2007) · Zbl 1129.30034 |
| [8] | De Bie, H.; Sommen, F., Fundamental solutions for the super Laplace and Dirac operators and all their natural powers, J. Math. Anal. Appl., 338, 1320-1328 (2008) · Zbl 1143.30022 |
| [9] | De Bie, H.; Sommen, F., Hermite and Gegenbauer polynomials in superspace using Clifford analysis, J. Phys. A, 40, 10441-10456 (2007) · Zbl 1152.30052 |
| [10] | H. De Bie, Schrödinger equation with delta potential in superspace, Phys. Lett. A, doi:10.1016/j.physleta.2008.04.005, in press; H. De Bie, Schrödinger equation with delta potential in superspace, Phys. Lett. A, doi:10.1016/j.physleta.2008.04.005, in press · Zbl 1221.81050 |
| [11] | De Bie, H.; Sommen, F., Spherical harmonics and integration in superspace, J. Phys. A, 40, 7193-7212 (2007) · Zbl 1143.30315 |
| [12] | Deans, S. R., The Radon Transform and Some of Its Applications (1983), Wiley-Interscience Publication, John Wiley & Sons Inc.: Wiley-Interscience Publication, John Wiley & Sons Inc. New York · Zbl 0561.44001 |
| [13] | Delanghe, R.; Sommen, F.; Souček, V., Clifford Algebra and Spinor-Valued Functions, Math. Appl., vol. 53 (1992), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0747.53001 |
| [14] | DeWitt, B., Supermanifolds, Cambridge Monogr. Math. Phys. (1984), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0551.53002 |
| [15] | Gelfand, I. M.; Shilov, G. E., Generalized Functions, vol. 1. Properties and Operations (1964), Academic Press [Harcourt Brace Jovanovich Publishers]: Academic Press [Harcourt Brace Jovanovich Publishers] New York, [1977] · Zbl 0115.33101 |
| [16] | Helgason, S., The Radon Transform, Progr. Math., vol. 5 (1999), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0932.43011 |
| [17] | Inoue, A., Foundation of real analysis on the superspace \(R^{m | n}\) over the ∞-dimensional Fréchet-Grassmann algebra, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39, 3, 419-474 (1992) · Zbl 0777.58005 |
| [18] | Inoue, A.; Maeda, Y., Foundations of calculus on super Euclidean space \(R^{m | n}\) based on a Fréchet-Grassmann algebra, Kodai Math. J., 14, 1, 72-112 (1991) · Zbl 0737.58009 |
| [19] | Kostant, B., Graded manifolds, graded Lie theory, and prequantization, (Differential Geometrical Methods in Mathematical Physics, Proc. Sympos.. Differential Geometrical Methods in Mathematical Physics, Proc. Sympos., Univ. Bonn, Bonn, 1975. Differential Geometrical Methods in Mathematical Physics, Proc. Sympos.. Differential Geometrical Methods in Mathematical Physics, Proc. Sympos., Univ. Bonn, Bonn, 1975, Lecture Notes in Math., vol. 570 (1977), Springer: Springer Berlin), 177-306 · Zbl 0358.53024 |
| [20] | Leĭtes, D. A., Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk, 35, 1(211), 3-57 (1980), 255 · Zbl 0439.58007 |
| [21] | Ozaktas, H.; Zalevsky, Z.; Kutay, M., The Fractional Fourier Transform (2001), Wiley: Wiley Chichester |
| [22] | Rogers, A., A global theory of supermanifolds, J. Math. Phys., 21, 6, 1352-1365 (1980) · Zbl 0447.58003 |
| [23] | Rogers, A., Stochastic calculus in superspace. I. Supersymmetric Hamiltonians, J. Phys. A, 25, 2, 447-468 (1992) · Zbl 0748.60053 |
| [24] | Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser., vol. 32 (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0232.42007 |
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.
