Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 047, 30 pages      arXiv:1105.0585      https://doi.org/10.3842/SIGMA.2011.047
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

The Fourier Transform on Quantum Euclidean Space

Kevin Coulembier
Gent University, Galglaan 2, 9000 Gent, Belgium

Received November 19, 2010, in final form April 21, 2011; Published online May 11, 2011

Abstract
We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's relations and a new type of q-Hankel transforms using the first and second q-Bessel functions. The behavior of the Fourier transforms with respect to partial derivatives and multiplication with variables is studied. The Fourier transform acts between the two representation spaces for the harmonic oscillator on quantum Euclidean space. By using this property it is possible to define a Fourier transform on the entire Hilbert space of the harmonic oscillator, which is its own inverse and satisfies the Parseval theorem.

Key words: quantum Euclidean space; Fourier transform; q-Hankel transform; harmonic analysis; q-polynomials; harmonic oscillator.

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References

  1. Atakishiyeva M.K., Atakishiyev N.M., q-Laguerre and Wall polynomials are related by the Fourier-Gauss transform, J. Phys. A: Math. Gen. 30 (1997), L429-L432.
  2. Bettaibi N., Bouzeffour F., Ben Elmonser H., Binous W., Elements of harmonic analysis related to the third basic zero order Bessel function, J. Math. Anal. Appl. 342 (2008), 1203-1219.
  3. Brzezinski T., Majid S., Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993), 591-638, Erratum, Comm. Math. Phys. 167 (1995), 235, hep-th/9208007.
  4. Carnovale G., On the braided Fourier transform on the n-dimensional quantum space, J. Math. Phys. 40 (1999), 5972-5997, math.QA/9810011.
  5. Carow-Watamura U., Schlieker M., Watamura S., SOq(N) covariant differential calculus on quantum space and quantum deformation of Schrödinger equation, Z. Phys. C 49 (1991), 439-446.
  6. Carow-Watamura U., Schlieker M., Watamura S., Weich W., Bicovariant differential calculus on quantum groups SUq(N) and SOq(N), Comm. Math. Phys. 142 (1991), 605-641.
  7. Carow-Watamura U., Watamura S., The q-deformed Schrödinger equation of the harmonic oscillator on the quantum Euclidean space, Internat. J. Modern Phys. A 9 (1994), 3989-4008.
  8. Ciccoli N., Koelink E., Koornwinder T., q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations, Methods Appl. Anal. 6 (1999), 109-127, math.CA/9805023.
  9. Coulembier K., De Bie H., Sommen F., Integration in superspace using distribution theory, J. Phys. A: Math. Theor. 42 (2009), 395206, 23 pages, arXiv:0909.2544.
  10. Coulembier K., Sommen F., q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials, J. Phys. A: Math. Theor. 43 (2010), 115202, 28 pages, arXiv:1002.4987.
  11. Coulembier K., De Bie H., Sommen F., Orthogonality of Hermite polynomials in superspace and Mehler type formulae, arXiv:1002.1118.
  12. De Bie H., Sommen F., Spherical harmonics and integration in superspace, J. Phys. A: Math. Theor. 40 (2007), 7193-7212, arXiv:0705.3148.
  13. Dunne R., Macfarlane A.J., de Azcárraga J.A., Pérez Bueno J.C., Supersymmetry from a braided point of view, Phys. Lett. B 387 (1996), 294-299, hep-th/9607220.
  14. Fiore G., The SOq(N,R)-symmetric harmonic oscillator on the quantum Euclidean space RqN and its Hilbert space structure, Internat. J. Modern Phys. A 8 (1993), 4679-4729, hep-th/9306030.
  15. Fiore G., Realization of Uq(so(N)) within the differential algebra on RqN, Comm. Math. Phys. 169 (1995), 475-500, hep-th/9403033.
  16. Floreanini R., Vinet L., Quantum algebra approach to q-Gegenbauer polynomials, J. Comput. Appl. Math. 57 (1995), 123-133.
  17. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge, Cambridge University Press, 1990.
  18. Hochstadt H., The functions of mathematical physics, Pure and Applied Mathematics, Vol. XXIII, Wiley-Interscience, New York - London - Sydney, 1971.
  19. Howe R., Tan E.C., Nonabelian harmonic analysis. Applications of SL(2,R), Universitext, Springer-Verlag, New York, 1992.
  20. Iorgov N., Klimyk A., Harmonics on the quantum Euclidean space, J. Phys. A: Math. Gen. 36 (2003), 7545-7558, math.QA/0302119.
  21. Ismail M.E.H., The zeros of basic Bessel functions, the functions Jν+ax(x), and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1-19.
  22. Jackson F., On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908), 253-281.
  23. Kempf A., Majid S., Algebraic q-integration and Fourier theory on quantum and braided spaces, J. Math. Phys. 35 (1994), 6802-6837.
  24. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  25. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Department of Technical Mathematics and Informatics, Delft University of Technology, Report no. 98-17, 1998, available at http://aw.twi.tudelft.nl/~koekoek/askey/.
  26. Koornwinder T., Special functions and q-commuting variables, in Special Functions, q-Series and Related Topics (Toronto, ON, 1995), Fields Inst. Commun., Vol. 14, Amer. Math. Soc., Providence, RI, 1997, 131-166, q-alg/9608008.
  27. Koornwinder T., Swarttouw R., On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), 445-461.
  28. Majid S., C-statistical quantum groups and Weyl algebras, J. Math. Phys. 33 (1992), 3431-3444.
  29. Majid S., Free braided differential calculus, braided binomial theorem, and the braided exponential map, J. Math. Phys. 34 (1993), 4843-4856, hep-th/9302076.
  30. Moak D., The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20-47.
  31. Ogievetsky O., Zumino B., Reality in the differential calculus on q-Euclidean spaces, Lett. Math. Phys. 25 (1992), 121-130, hep-th/9205003.
  32. Schirrmacher A., Generalized q-exponentials related to orthogonal quantum groups and Fourier transformations of noncommutative spaces, J. Math. Phys. 36 (1995), 1531-1546, hep-th/9409132.
  33. Sugitani T., Harmonic analysis on quantum spheres associated with representations of Uq(soN) and q-Jacobi polynomials, Compositio Math. 99 (1995), 249-281.
  34. Steinacker H., Integration on quantum Euclidean space and sphere, J. Math. Phys. 37 (1996), 4738-4749, q-alg/9506020.
  35. Wachter H., q-integration on quantum spaces, Eur. Phys. J. C Part. Fields 32 (2003), 281-297, hep-th/0206083.
  36. Wachter H., q-exponentials on quantum spaces, Eur. Phys. J. C Part. Fields 37 (2004), 379-389, hep-th/0401113.

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