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

SIGMA 5 (2009), 025, 15 pages      arXiv:0903.0573      https://doi.org/10.3842/SIGMA.2009.025
Contribution to the Special Issue on Dunkl Operators and Related Topics

Inversion Formulas for the Spherical Radon-Dunkl Transform

Zhongkai Li and Futao Song
Department of Mathematics, Capital Normal University, Beijing 100048, China

Received October 18, 2008, in final form March 01, 2009; Published online March 03, 2009

Abstract
The spherical Radon-Dunkl transform Rκ, associated to weight functions invariant under a finite reflection group, is introduced, and some elementary properties are obtained in terms of h-harmonics. Several inversion formulas of Rκ are given with the aid of spherical Riesz-Dunkl potentials, the Dunkl operators, and some appropriate wavelet transforms.

Key words: spherical Radon-Dunkl transform; h-harmonics; inversion formula; wavelet.

pdf (295 kb)   ps (199 kb)   tex (18 kb)

References

  1. Campi S., On the reconstruction of a function on a sphere by its integrals over great circles, Boll. Un. Mat. Ital. C (5) 18 (1981), 195-215.
  2. Dai F., Xu Y., Maximal function and multiplier theorem for weighted space on the unit sphere, J. Funct. Anal. 249 (2007), 477-504, math.CA/0703928.
  3. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
  4. Dunkl C.F., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
  5. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  6. Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227.
  7. Dunkl C.F., Hankel transforms associated to finite reflection groups, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials and Applications (Tampa, 1991), Contemp. Math. 138 (1992), 123-138.
  8. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  9. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vols. I and II, McGraw-Hill Book Company, Inc., New York - Toronto - London, 1953, 1953.
  10. Gindikin S., Reeds J., Shepp L., Spherical tomography and spherical integral geometry, in Tomography, Impedance Imaging, and Integral Geometry (South Hadley, MA, 1993), Lectures in Appl. Math., Vol. 30, Amer. Math. Soc., Providence, RI, 1994, 83-92.
  11. Goodey P., Weil W., Centrally symmetric convex bodies and the spherical Radon transform, J. Differential Geom. 35 (1992), 675-688.
  12. Gradshteyn I.S., Ryzhik L.M., Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000.
  13. Grinberg E.L., Spherical harmonics and integral geometry on projective spaces, Trans. Amer. Math. Soc. 279 (1983), 187-203.
  14. Guillemin V., Radon transform on Zoll surfaces, Adv. in Math. 22 (1976), 85-119.
  15. Helgason S., The Radon transform, 2nd ed., Progress in Mathematics, Vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999.
  16. Helgason S., Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, Vol. 39, American Mathematical Society, Providence, RI, 1994.
  17. Kurusa A., The Radon transform on half sphere, Acta Sci. Math. (Szeged) 58 (1993), 143-158.
  18. Rösler M., Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463, q-alg/9710029.
  19. Rösler M., A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), 2413-2438, math.CA/0210137.
  20. Rubin B., Fractional integrals and potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 82, Longman, Harlow, 1996.
  21. Rubin B., Spherical Radon transform and related wavelet transforms, Appl. Comput. Harmon. Anal. 5 (1998), 202-215.
  22. Rubin B., Inversion of fractional integrals related to spherical Radon transform, J. Funct. Anal. 157 (1998), 470-487.
  23. Rubin B., Fractional integrals and wavelet transforms associated with Blaschke-Levy representations on the sphere, Israel J. Math. 114 (1999), 1-27.
  24. Rubin B., Inversion and characterization of the hemiopherical transform, J. Anal. Math. 77 (1999), 105-128.
  25. Rubin B., Generalized Minkowski-Funk transforms and small denominators on the sphere, Fract. Calc. Appl. Anal. 3 (2000), 177-203.
  26. Rubin B., Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. in Appl. Math. 29 (2002), 471-497.
  27. Rubin B., Ryabogin D., The k-dimensional Radon transform on the n-sphere and related wavelet transforms, in Radon Transforms and Tomography (South Hadley, MA, 2000), Contemp. Math. 278 (2001), 227-239.
  28. Strichartz R.S., Lp estimates for Radon transforms in Euclidean and non-Euclidean spaces, Duke Math. J. 48 (1981), 699-727.
  29. Trimèche K., The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transforms Spec. Funct. 12 (2001), 349-374.
  30. Trimèche K., Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct. 13 (2002), 17-38.
  31. Xu Y., Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.
  32. Xu Y., Intertwining operator and h-harmonic associated with reflection groups, Canad. J. Math. 50 (1998), 193-209.
  33. Xu Y., Approximation by means of h-harmonic polynomials on the unit sphere, Adv. Comput. Math. 21 (2004), 37-58.
  34. Xu Y., Weighted approximation of functions on the unit sphere, Constr. Approx. 21 (2005), 1-28, math.CA/0312525.
  35. Xu Y., Almost everywhere convergebce of orthogonal expansions of several variables, Constr. Approx. 22 (2005), 67-93, math.CA/0312526.
  36. Xu Y., Generalized translation operator and approximation in several variables, J. Comput. Appl. Math. 178 (2005), 489-512, math.CA/0401417.

Previous article   Next article   Contents of Volume 5 (2009)