Polynomial Harmonic Decompositions

For real polynomials in two indeterminates a classical polynomial harmonic decomposition (cf. (1) below) is extended from square-norm divisors to conic ones. The main result is then applied to obtain a full polynomial harmonic decomposition, and to solve a Dirichlet problem with polynomial boundary data. Harmonic functions are of utmost importance in analysis, geometry, and mathematical physics [1]. Even at their most basic occurrence, as polynomial harmonic functions, they produce surprisingly useful results. One of them is the following classical harmonic decomposition: In R, n ≥ 2, with coordinates x = (x1, x2, . . . , xn), any homogeneous real polynomial of degree m ≥ 0, pm(x), is uniquely decomposable as pm(x) = hm(x) + |x|pm−2(x), (1) where hm(x) is a homogeneous harmonic (∑ j ∂hm ∂xj = 0 ) real polynomial of degree m, |x| = ∑ j x 2 j , and pm−2(x) is a homogeneous polynomial of degree m − 2 (possibly 0). As customary, here and in what follows the concepts of polynomial and polynomial function will be identified and used interchangeably. There are vast generalizations of the decomposition (1), where the harmonic polynomials are replaced by polynomial solutions to constant coefficient partial differential operators, and |x| by more general polynomials, like the symbols of those operators [4, 5]. Key applications of the decomposition (1)