Solution of the Dirichlet Problem by Interpolating Harmonic Polynomials

1. Introduction. Let Dbea bounded simply connected region of the complex z-plane which is regular for the Dirichlet problem, let C be the boundary of D, and let u be a continuous function on C to the real numbers. Some time ago J. L. Walsh [l, p. 517] suggested that it might be possible to define a sequence of harmonic polynomials by coincidence with the values of u in points so chosen on C that the polynomials converge on D to the solution of the corresponding Dirichlet problem. He showed that a convergent interpolation process of this type is available if D is a circular disk [l; 2]. Recently the author [3] raised the general question anew, and Walsh [4] then extended his earlier result to elliptical disks. For both theoretical and practical reasons it seems worthwhile to develop a general convergence theory, and progress toward this goal is announced here in Theorems 3.2 and 3.3 below.