The discrete analog of a minimum problem in conformal mapping

Abstract

A discrete analog is given to a well-known minimum problem in the Hilbert space of square-integrable analytic functions-on a bounded domain. The problem is to find the function with minimum norm from among all functions in the space having the value 1 at a given fixed point. J. Ferrand, R. J. Duffin, and others have developed the theory of discrete analytic (preholomorphic} functions. For the discrete analog of the minimum problem, a subset of the set of discrete analytic functions defined on a square net contained in the domain is considered. A discrete minimum problem is stated and the solution of this problem can be expressed as the solution of a system of equations. It is shown that, with certain conditions on the boundary of the domain and the fixed point, the solution of the discrete minimum problem converges to the solution of the ordinary minimum problem as the net width decreases to zero.