Analytic continuation from data points with unequal errors

The problem solved in this paper is that of constructing zero-free holomorphic functions which will (a) assume specified values at a set of discrete data points in a data region Γ1 inside the holomorphy domain, and which at the same time will (b) provide the optimum solution to various stabilizing (boundedness or smoothness) conditions on the boundaries ΓR. The immediate motivation to this problem arose from the need to renormalize data with unequal errors by using a holomorphic weight function to bring all the errors to the same value: this was a preliminary step before making an analytic continuation off the data region Γ1. Since a stabilizing condition has to be imposed on the boundaries ΓR, the weight function must be chosen so as to introduce the minimum additional instability on ΓR. Although this was the specific motivation, other interesting applications suggest themselves and some of these are discussed. The stability conditions on ΓR which are treated may all be expressed in terms of the real parts, or of the normal derivative of the real parts. Γ1 is taken to be on the real axis and the functions considered satisfy a reflection principle which means that the data values are real. It follows that the results obtained may be expressed in terms of the real parts alone—in other words the problem solved here is in fact that of obtaining harmonic functions which take specific values inside their harmonicity domain and which satisfy the appropriate extremum condition on the boundary.