An integral expression for the Dunkl kernel in the dihedral setting

• Autorzy: Maslouhi M.

Identyfikatory

DOI

10.1515/jaa-2017-0011

• Języki publikacji: EN

Abstrakty

EN

Using the results in [12] where a construction of the Dunkl intertwining operator for a large set of regular parameter functions is provided, we establish an integral expression for the Dunkl kernel in the context of the dihedral group D_n with constant parameter function k ∈ ℂ and arbitrary order n ≥ 2. Our main tool is a differential system that leads to the explicit expression of the Dunkl kernel whenever an appropriate solution of it is obtained. In particular, an explicit expression of the Dunkl kernel E_k(x, y) is given when one of its argument x or y is invariant under the action of any reflection in the dihedral group. We obtain also a generating series for the homogeneous components K_m(x, y), m ∈ ℤ⁺, of the Dunkl kernel and provide new sharp estimates for the Dunkl kernel in the large context k ∈ ℂ, n ≥ 2 and −2nk ≠ 1, 2, 3, . . . .

Słowa kluczowe

EN

dihedral group; Dunkl operator; Dunkl kernel; Dunkl intertwining operator

PL

grupa diedralna; operator Dunkla; jądro Dunkla

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2017
• Tom: Vol. 23, nr 2
• Strony: 73--84
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Maslouhi M.

• National School of Applied Sciences, Ibn Tofail University, 14000 Kenitra, Morocco

Bibliografia

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• [6] C. F. Dunkl, Hankel transforms associated to finite reflection groups, in: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa 1991), Contemp. Math. 138, American Mathematical Society, Providence (1992), 123-138.
• [7] C. F. Dunkl, M. F. E. de Jeu and E. M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), no. 1, 237-256.
• [8] C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia Math. Appl. 81, Cambridge University Press, Cambridge, 2001.
• [9] C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, 2nd ed., Encyclopedia Math. Appl. 155, Cambridge University Press, Cambridge, 2014.
• [10] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge University Press, Cambridge, 1990.
• [11] L. Lapointe and L. Vinet, Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), no. 2, 425-452.
• [12] M. Maslouhi and E. H. Youssfi, The Dunkl intertwining operator, J. Funct. Anal. 256 (2009), no. 8, 2697-2709.
• [13] M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), no. 3, 445-463.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).