Harmonic Polynomials and
Dirichlet-Type Problems

Sheldon Axler and Wade Ramey

This paper was published in the Proceedings of the American Mathematical Society 123 (1995), 3765-3773.

Abstract: Surprisingly powerful results about harmonic functions can be obtained simply by differentiating the function |x|2-n and observing the patterns that emerge. This theme leads to an explicit formula for a natural projection of a polynomial into the space of harmonic polynomials. The ease with which this projection can be calculated gives rise to a fast algorithm for computing the Poisson integral of any polynomial. (Note: The algorithm involves differentiation, but no integration.) We show how this algorithm can be used for many other Dirichlet-type problems with polynomial data, including the Neumann problem, the exterior Dirichlet problem, the annular Dirichlet problem, the Bergman projection problem, and the biDirichlet problem. Finally, we give a new and simple proof that the Kelvin transform preserves harmonic functions.

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