On the Degree of Convergence of Harmonic Polynomials to Harmonic Functions.

The purpose of this note is to state without proof certain. results on polynomial approximation to harmonic functions whose proofs will appear elsewhere later. In the plane of the complex variable z = x + iy, a (real) function u(z) harmonic in the interior of the Jordan curve C and continuous in the corresponding closed region C can be represented in C by a uniformly convergent series of harmonic polynomials.1 If CR denotes the image of the circle w = R > 1 under the conformal map of the exterior of C onto the region w > 1 of the w-plane so that the points at infinity in the two planes correspond to each other, and if u(z) is harmonic throughout the interior of CR, then there exist harmonic polynomials pn(z) of respective degrees n = 1, 2, ..., such that lim sup [max. Iu(z) p.(z) on C] iln < 1/R; (1)