Clifford-Hermite-monogenic operators |
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Authors: | Fred Brackx Nele de Schepper Frank Sommen |
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Institution: | (1) Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, B-9000 Ghent, Belgium;(2) Clifford Research Group, Department of Mathematical Analysis, Faculty of Sciences, Ghent University, Galglaan 2, B-9000 Ghent, Belgium |
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Abstract: | In this paper we consider operators acting on a subspace ℳ of the space L
2 (ℝm; ℂm) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace
ℳ is defined as the orthogonal sum of spaces ℳs,k of specific Clifford basis functions of L
2(ℝm; ℂm).
Every Clifford endomorphism of 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 ℳs,k into a similar space ℳs′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known.
Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is
studied in detail. |
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Keywords: | differential operators Clifford analysis |
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