Generalized Almansi Expansions in Superspace
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Generalized Almansi Expansions in Superspace Hongfen Yuan1
Received: 15 December 2014 / Revised: 2 November 2015 / Accepted: 5 December 2015 © Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, we first study an expansion for the operators (∂x − λ)k , where ∂x is the Dirac operator in superspace and λ is a complex number. Then we investigate expansions for polynomial Dirac operators in superspace. These expansions are regarded as generalized Almansi expansions in superspace. As an application of the expansions, the modified Riquier problem in superspace is considered. Keywords Superspace · Polynomial Dirac operator · Almansi expansion · Riquier problem Mathematics Subject Classification
Primary 30G35; Secondary 58C50
1 Introduction One of the main aims of Clifford analysis is to study the function-theoretical properties of the null-solutions of the Dirac operator generalizing the Cauchy–Riemann
Communicated by Klaus Gürlebeck. Research supported by the TianYuan Special Funds of the National Natural Science Foundation of China under Grant No. 11426082, and Project of Handan Municipal Science and Technology Bureau under Grant No. 1534201097-10.
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Hongfen Yuan [email protected] College of Science, Hebei University of Engineering, Handan 056038, China
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H. Yuan
operator in higher-dimensional spaces (see [1]). More recently, Clifford analysis in superspace has emerged as a new branch of the Clifford analysis. Superspaces, developed during the second half of the previous century, are spaces equipped with both a set of commuting variables and a set of anti-commuting variables (generating the so-called Grassmann algebra) to describe the properties of bosons and fermions in Quantum Mechanics. F. Sommen, H. DeBie and others studied a superspace of dimension (m, 2n) in the frame of Clifford analysis. They studied the functions defined in the superspace Rm|2n and taking values in the Clifford algebra (including orthogonal and symplectic Clifford algebras). Then they defined generalized differential operators acting on these functions, such as a super-Dirac operator, etc. (see [2]). Furthermore, the Fischer decomposition in superspace is given in [3], which is the direct-sum decomposition of the polynomial algebra generated by the commuting and anti-commuting variables into products of spaces of spherical harmonics with powers of a generalized norm squared of a vector. Based on the work of Sommen et al., we investigated the Almansi expansion in superspace. Corresponding to the development of Clifford analysis, a number of mathematicians have made great progress in Almansi expansions. Without claiming completeness, we mention some of them. In 1899, the Almansi expansion for polyharmonic functions was established, which was equivalent to the Fischer decomposition for polynomials (see [4]). Indeed the expansion builds the relation between harmonic functions and polyharmonic functions, which plays a central role in the theory of polyharmonic functions. The result in the case of harmonic analysis, complex an
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