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On homogeneous polynomial solutions of the Riesz system and their harmonic potentials. (English) Zbl 1201.30063

Summary: The spaces \(M^+(\mathbb R^{m+1}; \mathbb R^{0,m+1}; k)\), \(k\in \mathbb N\), of vector-valued homogeneous monogenic polynomials of degree \(k\) in \(\mathbb R^{m+1}\) are locally the building blocks of solutions to the Riesz system in \(\mathbb R^{m+1}\). Furthermore, the Dirac operator \(\partial_x\) in \(\mathbb R^{m+1}\) determines an isomorphism between \(\mathcal H(\mathbb R^{m+1}; k+1)\) – the space of solid harmonics of degree \((k+1)\) in \(\mathbb R^{m+1}\) – and \(M^+(\mathbb R^{m+1}; \mathbb R^{0,m+1}; k)\). Relying essentially upon primitivation, a step-by-step procedure for constructing bases of these spaces is established (section 2). Particular bases in terms of the Fueter polynomials and their harmonic potentials are given in section 3.

MSC:

30G35 Functions of hypercomplex variables and generalized variables
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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