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Keywords:
Dunkl Laplacian; reproducing kernel
Dunkl Laplacian; reproducing kernel
Summary:
In this paper, we compute explicitly the reproducing kernel of the space of homogeneous polynomials of degree $n$ and Dunkl polyharmonic of degree $m$, i.e. $\Delta_{k}^{m}u=0$, $m\in \mathbb{N}\setminus\{0\}$, where $\Delta_{k}$ is the Dunkl Laplacian and we study the convergence of the orthogonal series of Dunkl polyharmonic homogeneous polynomials.
References:
[1] Dunkl C.F.: Differential-difference operators associated to reflection group. Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183. DOI 10.1090/S0002-9947-1989-0951883-8 | MR 0951883
[2] Dunkl C.F., Xu Y.: Orthogonal Polynomials of Several Variables. Cambridge Univ. Press, Cambridge, 2001. MR 1827871 | Zbl 0964.33001
[3] Kuran Ü.: On Brelot-Choquet axial polynomials. J. London Math. Soc. (2) 4 (1971), 15–26. DOI 10.1112/jlms/s2-4.1.15 | MR 0293116 | Zbl 0219.31013
[4] Mejjaoli H., Trimèche K.: On a mean value property associated with the Dunkl Laplacian operator and applications. Integral Transform. Spec. Funct. 12 (2001), no. 3, 279–302. DOI 10.1080/10652460108819351 | MR 1872437
[5] Ren G.B.: Almansi decomposition for Dunkl operators. Sci. China Ser. A 48 (2005), suppl., 333–342. DOI 10.1007/BF02884718 | MR 2156514 | Zbl 1131.43010
[6] Render H.: Reproducing kernels for polyharmonic polynomials. Arch. Math. 91 (2008), 136–144. DOI 10.1007/s00013-008-2447-9 | MR 2430797 | Zbl 1151.31007
[7] Rösler M.: Dunkl operators: theory and applications. Orthogonal polynomials and special functions. (Leuven, 2002), Lecture Notes in Mathematics, 1817, Springer, Berlin, 2003, pp. 93–135. DOI 10.1007/3-540-44945-0_3 | MR 2022853
[8] Rösler M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192 (1998), 519–542. DOI 10.1007/s002200050307 | MR 1620515
[9] Trimèche K.: The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual. Integral Transform. Spec. Funct. 12 (2001), no. 4, 349–374. DOI 10.1080/10652460108819358 | MR 1872375
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