We prove some results about the Fueter-regular homogeneous polynomials, which appear as components in the power series of any quaternionic regular function. Let B denote the unit ball in C2 ≅ H and S = δ B the group of unit quaternions. In §2.1 we obtain a differential condition that characterize the homogeneous polynomials whose restrictions to S extend as a regular polynomial. This result generalizes a similar characterization for holomorphic extensions of polynomials proved by Kytmanov. In §2.2 we show how to define an injective linear operator R: Hk(S)-> Uk Ψ between the space of complex-valued spherical harmonics and the H-module of regular homogeneous polynomials of degree k. In particular, we show how to construct bases of the module of regular homogeneous polynomials of a fixed degree starting from any choice of C-bases of the spaces of complex harmonic homogeneous polynomials.
On regular harmonics of one quaternionic variable / Perotti, Alessandro. - ELETTRONICO. - (2005), pp. 1-7.
On regular harmonics of one quaternionic variable
Perotti, Alessandro
2005-01-01
Abstract
We prove some results about the Fueter-regular homogeneous polynomials, which appear as components in the power series of any quaternionic regular function. Let B denote the unit ball in C2 ≅ H and S = δ B the group of unit quaternions. In §2.1 we obtain a differential condition that characterize the homogeneous polynomials whose restrictions to S extend as a regular polynomial. This result generalizes a similar characterization for holomorphic extensions of polynomials proved by Kytmanov. In §2.2 we show how to define an injective linear operator R: Hk(S)-> Uk Ψ between the space of complex-valued spherical harmonics and the H-module of regular homogeneous polynomials of degree k. In particular, we show how to construct bases of the module of regular homogeneous polynomials of a fixed degree starting from any choice of C-bases of the spaces of complex harmonic homogeneous polynomials.File | Dimensione | Formato | |
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