Generalization of Padé approximation from rational functions to arbitrary analytic functions — Theory

Abstract:

The Padé approximation has a long and rich history of theory and application and is known to produce excellent local approximations. We present a method for extending the basic idea of Padé approximation, that of matching a prescribed number of terms in the Taylor series expansion of a given function using rational functions, to any arbitrary function holormorphic in a neighborhood of the Taylor series expansion point. We demonstrate that providing the flexibility of using other functions in a Padé-type approximation yields highly accurate approximations having additional desirable properties. Additional properties that can be preserved in our method include comparable asymptotic behavior to the function to be approximated, or the preservation of band-limitedness in the approximation.