Some Problems in Numerical Analytic Continuation

Abstract

of a Riemann sum approximation to the Cauchy integral formula by a linear programming problem. The method is quite general, although the estimates are valid only for the disc. Recently, Henrici [4] has proposed an algorithm for analytic continuation in a general domain from exact values for a finite number of derivatives of the function at a point. This is based upon the Weierstrass circle-chain method. Miller [6] discusses two methods for analytic continuation on a disc; one method involves a truncated Fourier expansion when data are given on an entire interior circle; the other in? volves a linear programming problem when data are given at the vertices of a polygon enclosing an interior disc. In the present paper the authors consider three problems in a general Jordan domain. In each case, an ap? proximating polynomial is determined by a linear programming problem. The existence of such a polynomial is insured by a theorem of Walsh [7]. Specifically, let D be a Jordan domain in the complex plane. Let f(z) denote an unknown function analytic in D and continuous on D. Problems of analytic continuation fall into the large class of improperly posed prob? lems for which continuous dependence of the solution on the data can be restored by restricting attention to those solutions satisfying a prescribed bound. For examples, see [5]. We therefore assume always that/(z) satisfies the prescribed bound