| Title:
|
Clifford-Hermite-monogenic operators (English) |
| Author:
|
Brackx, Fred |
| Author:
|
de Schepper, Nele |
| Author:
|
Sommen, Frank |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
1301-1322 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider operators acting on a subspace $\mathcal M$ of the space $L_2(\mathbb{R}^m;\mathbb{C}_m)$ of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ${\mathcal M}$ is defined as the orthogonal sum of spaces ${\mathcal M}_{s,k}$ of specific Clifford basis functions of $L_2(\mathbb{R}^m;\mathbb{C}_m)$. Every Clifford endomorphism of ${\mathcal M}$ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ${\mathcal M}_{s,k}$ into a similar space ${\mathcal M}_{s^{\prime }\!,k^{\prime }}$. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ${\mathcal M}$ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail. (English) |
| Keyword:
|
differential operators |
| Keyword:
|
Clifford analysis |
| MSC:
|
30G35 |
| MSC:
|
47B99 |
| MSC:
|
47F05 |
| idZBL:
|
Zbl 1164.47336 |
| idMR:
|
MR2280810 |
| . |
| Date available:
|
2009-09-24T11:42:58Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128146 |
| . |
| Reference:
|
[1] F. Brackx, R. Delanghe, F. Sommen: Clifford Analysis.Pitman Publ., Boston-London-Melbourne, 1982. MR 0697564 |
| Reference:
|
[2] F. Brackx, N. De Schepper, K. I. Kou, and F. Sommen: The Mehler formula for the generalized Clifford-Hermite polynomials.Acta Mathematica Sinica, Accepted. |
| Reference:
|
[3] R. Delanghe, F. Sommen, and V. Souček: Clifford Algebra and Spinor-Valued Functions.Kluwer Acad. Publ., Dordrecht, 1992. MR 1169463 |
| Reference:
|
[4] F. Sommen: Special functions in Clifford analysis and axial symmetry.J. Math. Anal. Appl. 130 (1988), 110–133. Zbl 0634.30042, MR 0926831, 10.1016/0022-247X(88)90389-7 |
| Reference:
|
[5] F. Sommen, N. Van Acker: Monogenic differential operators.Results Math. 22 (1992), 781–798. MR 1189765, 10.1007/BF03323123 |
| . |
