An integral expression for the Dunkl kernel in the dihedral setting. (English) Zbl 06815248
Summary: 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 \(\mathcal{D}_n\) with constant parameter function \(k\in\mathbb{C}\) and arbitrary order \(n\geq2\). 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 \(K_m(x,y)\), \(m\in\mathbb{Z}^+\), of the Dunkl kernel and provide new sharp estimates for the Dunkl kernel in the large context \(k\in\mathbb{C}\), \(n\geq2\) and \(-2nk\neq 1,2,3,\dots\).
MSC:
| 47B48 | Linear operators on Banach algebras |
| 33C52 | Orthogonal polynomials and functions associated with root systems |
| 33C67 | Hypergeometric functions associated with root systems |
| 31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |
References:
| [1] | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1966.; Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (1966) · Zbl 0515.33001 |
| [2] | M. de Jeu, Paley-Wiener theorems for the Dunkl transform, Trans. Amer. Math. Soc. 358 (2006), no. 10, 4225-4250.; de Jeu, M., Paley-Wiener theorems for the Dunkl transform, Trans. Amer. Math. Soc., 358, 10, 4225-4250 (2006) · Zbl 1160.33010 |
| [3] | M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147-162.; de Jeu, M. F. E., The Dunkl transform, Invent. Math., 113, 1, 147-162 (1993) · Zbl 0789.33007 |
| [4] | C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167-183.; Dunkl, C. F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc., 311, 1, 167-183 (1989) · Zbl 0652.33004 |
| [5] | C. F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), no. 6, 1213-1227.; Dunkl, C. F., Integral kernels with reflection group invariance, Canad. J. Math., 43, 6, 1213-1227 (1991) · Zbl 0827.33010 |
| [6] | C. F. Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa 1991), Contemp. Math. 138, American Mathematical Society, Providence (1992), 123-138.; Dunkl, C. F., Hankel transforms associated to finite reflection groups, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, 123-138 (1992) · Zbl 0789.33008 |
| [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.; Dunkl, C. F.; de Jeu, M. F. E.; Opdam, E. M., Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc., 346, 1, 237-256 (1994) · Zbl 0829.33010 |
| [8] | C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia Math. Appl. 81, Cambridge University Press, Cambridge, 2001.; Dunkl, C. F.; Xu, Y., Orthogonal Polynomials of Several Variables (2001) · Zbl 0964.33001 |
| [9] | C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, 2nd ed., Encyclopedia Math. Appl. 155, Cambridge University Press, Cambridge, 2014.; Dunkl, C. F.; Xu, Y., Orthogonal Polynomials of Several Variables (2014) · Zbl 1317.33001 |
| [10] | J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge University Press, Cambridge, 1990.; Humphreys, J. E., Reflection Groups and Coxeter Groups (1990) · Zbl 0725.20028 |
| [11] | L. Lapointe and L. Vinet, Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), no. 2, 425-452.; Lapointe, L.; Vinet, L., Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys., 178, 2, 425-452 (1996) · Zbl 0859.35103 |
| [12] | M. Maslouhi and E. H. Youssfi, The Dunkl intertwining operator, J. Funct. Anal. 256 (2009), no. 8, 2697-2709.; Maslouhi, M.; Youssfi, E. H., The Dunkl intertwining operator, J. Funct. Anal., 256, 8, 2697-2709 (2009) · Zbl 1175.33010 |
| [13] | M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), no. 3, 445-463.; Rösler, M., Positivity of Dunkl’s intertwining operator, Duke Math. J., 98, 3, 445-463 (1999) · Zbl 0947.33013 |
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