Extrapolation - Extrapolation in The Complex Plane

Extrapolation in The Complex Plane

In complex analysis, a problem of extrapolation may be converted into an interpolation problem by the change of variable . This transform exchanges the part of the complex plane inside the unit circle with the part of the complex plane outside of the unit circle. In particular, the compactification point at infinity is mapped to the origin and vice versa. Care must be taken with this transform however, since the original function may have had "features", for example poles and other singularities, at infinity that were not evident from the sampled data.

Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is expanded at one of its points of convergence to produce a power series with a larger radius of convergence. In effect, a set of data from a small region is used to extrapolate a function onto a larger region.

Again, analytic continuation can be thwarted by function features that were not evident from the initial data.

Also, one may use sequence transformations like Padé approximants and Levin-type sequence transformations as extrapolation methods that lead to a summation of power series that are divergent outside the original radius of convergence. In this case, one often obtains rational approximants.