Optimized data extrapolation off the boundary of an analytic function

Abstract

We discuss the problem of extrapolation of a function whose real and imaginary parts are not equally well known on the boundary of its analytic domain. Using an integral equation for the error correlation function, we examine the optimized extrapolated value of the function and the probable error in extrapolation. For several cases, specific solutions are obtained which illustrate general properties and are also useful in phenomenological applications. Comparing with ordinary dispersion relations, we note when the extrapolation error may be uncontrolled, or only weakly constrained, by the input data, and how a knowledge of other properties of the function; e.g. its threshold or high energy behavior, may be necessary in order to reduce the extrapolation error.