Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

THE CHEREDNIK AND THE GAUSSIAN CHEREDNIK WINDOWED TRANSFORMS ON ℝd IN THE W-INVARIANT CASE

  • Received : 2019.11.27
  • Accepted : 2020.09.18
  • Published : 2020.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we give the harmonic analysis associated with the Cherednik operators, next we define and study the Cherednik wavelets and the Cherednik windowed transforms on ℝd, in the W-invariant case, and we prove for these transforms Plancherel and inversion formulas. As application we give these results for the Gaussian Cherednik wavelets and the Gaussian Cherednik windowed transform on ℝd in the W-invariant case.

Keywords

References

  1. I.Cherednik, Inverse Harish-Chandra transform and difference operators, Internat. Math. Res. Notices 15 (1997), 733-750. https://doi.org/10.1155/S1073792897000482
  2. L.Gallardo and K.Trimeche, Positivity of the Jacobi-Cherednik intertwining operator and its dual, Adv. Pure Appl. Math. 1 (2012), 163-194. https://doi.org/10.1515/APAM.2010.011
  3. G.J.Heckman and E.M.Opdam, Root systems and hypergeometric functions I, Compositio Math. 64 (1987), 329-352.
  4. T.H.Koornwinder, A new proof of the Paley-Wiener type theorem for the Jacobi transform, Arkiv For Math. 13 (1) (1975), 145-159. https://doi.org/10.1007/BF02386203
  5. T.H.Koornwinder, The continuous wavelet transform. Series in Approximations and Decompositions. Vol. 1. Wavelets: An elementary treatment of theory and applications. Edited by T.H.Koornwinder, World Scientific, (1993), 27-48.
  6. E.M.Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121. https://doi.org/10.1007/BF02392487
  7. B.Schapira, Etude analytique et probabiliste de Laplacians associes a des systemes de racines: Laplacian hypergeometrique de Heckman-Opdam et Laplacian combinatoire sur les immeubles affines, Ph. D. Thesis, University of Orleans, 2006.
  8. B.Schapira, Contribution to the hypergeometric function theory of Heckman and Opdam; sharp estimates, Schwartz spaces, heat kernel, Geom. Funct. Anal. 18 (2008), 222-250. https://doi.org/10.1007/s00039-008-0658-7
  9. K.Trimeche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, 1997.
  10. K.Trimeche, The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operator and the Heckman Opdam theory, Adv. Pure Appl. Math. 1 (2010), 293-323. https://doi.org/10.1515/APAM.2010.015
  11. K.Trimeche, Harmonic analysis associated with the Cherednik operators and the Heckman-Opdam theory, Adv. Pure Appl. Math. 2 (2011), 23-46. https://doi.org/10.1515/APAM.2011.022
  12. K.Trimeche, The harmonic analysis associated to the Heckman-Opdam theory and its applications to the root system of type BCd, Korean J. Math. 27 (2019) (1), 221-267. https://doi.org/10.11568/KJM.2019.27.1.221