Harmonic polynomial

Summary

A harmonic polynomial is a polynomial in $n$ real variables whose Laplacian is the zero polynomial. Harmonic polynomials are the simplest harmonic functions on all of $\R^n$. The homogeneous ones of degree $k$ form a finite-dimensional space, and their restrictions to the unit sphere are the spherical harmonics. Every harmonic function on a ball is a locally uniform limit of harmonic polynomials, so they play the role that monomials play for power series.