Of Approximation to a Harmonic Function

THEOREM I. Let C be an arbitrary closed Jordan region of the (x, y)-plane, and let w = {z), z~x+iy, be a function which maps conformally the exterior of C onto the exterior of the unit circle in the w-plane so that the points at infinity correspond to each other. Let CR denote the curve \cf>(z) \=R, R>1, that is, the transform onto the z-plane of the circle \w\ = R. A necessary and sufficient condition that an arbitrary function u(x, y), defined in C, be harmonic in (the closed region) C is that there should exist harmonic polynomials pn(x, y) of degree n,"f n~0, 1, 2, • • • , and numbers M, R>1, such that the inequalities