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.


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