Clifford-Hermite-monogenic operators
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1301-1322
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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.
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.
@article{CMJ_2006_56_4_a15,
author = {Brackx, Fred and de Schepper, Nele and Sommen, Frank},
title = {Clifford-Hermite-monogenic operators},
journal = {Czechoslovak Mathematical Journal},
pages = {1301--1322},
year = {2006},
volume = {56},
number = {4},
mrnumber = {2280810},
zbl = {1164.47336},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a15/}
}
Brackx, Fred; de Schepper, Nele; Sommen, Frank. Clifford-Hermite-monogenic operators. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1301-1322. http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a15/
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