| Title:
|
Reproducing kernels for Dunkl polyharmonic polynomials (English) |
| Author:
|
Touahri, Kamel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
53 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
37-50 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Dunkl Laplacian |
| Keyword:
|
reproducing kernel |
| MSC:
|
31B30 |
| MSC:
|
33C55 |
| idZBL:
|
Zbl 1249.33011 |
| idMR:
|
MR2880909 |
| . |
| Date available:
|
2012-02-07T10:22:15Z |
| Last updated:
|
2014-04-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141824 |
| . |
| Reference:
|
[1] Dunkl C.F.: Differential-difference operators associated to reflection group.Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183. MR 0951883, 10.1090/S0002-9947-1989-0951883-8 |
| Reference:
|
[2] Dunkl C.F., Xu Y.: Orthogonal Polynomials of Several Variables.Cambridge Univ. Press, Cambridge, 2001. Zbl 0964.33001, MR 1827871 |
| Reference:
|
[3] Kuran Ü.: On Brelot-Choquet axial polynomials.J. London Math. Soc. (2) 4 (1971), 15–26. Zbl 0219.31013, MR 0293116, 10.1112/jlms/s2-4.1.15 |
| Reference:
|
[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. MR 1872437, 10.1080/10652460108819351 |
| Reference:
|
[5] Ren G.B.: Almansi decomposition for Dunkl operators.Sci. China Ser. A 48 (2005), suppl., 333–342. Zbl 1131.43010, MR 2156514, 10.1007/BF02884718 |
| Reference:
|
[6] Render H.: Reproducing kernels for polyharmonic polynomials.Arch. Math. 91 (2008), 136–144. Zbl 1151.31007, MR 2430797, 10.1007/s00013-008-2447-9 |
| Reference:
|
[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. MR 2022853, 10.1007/3-540-44945-0_3 |
| Reference:
|
[8] Rösler M.: Generalized Hermite polynomials and the heat equation for Dunkl operators.Comm. Math. Phys. 192 (1998), 519–542. MR 1620515, 10.1007/s002200050307 |
| Reference:
|
[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. MR 1872375, 10.1080/10652460108819358 |
| . |
