Interpolation of infinite sequences by entire functions

Abstract

The study was originally motivated by a theoretical question raised by Warming and Hyett in a famous publication on the Modified Equation Approach [R.F. Warming, B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput Phys. 14 (1974) 159–179]. Their classical accuracy analysis of a finite-difference method applied to a time-dependent problem implicitly relies on the assumption that a function interpolating the numerical values can be expanded, over an indefinite domain, in Taylor's series of the independent variables x and t. Here, we establish constructively that the problem of interpolation of an arbitrary infinite sequence of real numbers by an entire function of x (and possibly t) admits uncountably many solutions. In the case of a single variable, if the values are bounded, the interpolant can be made bounded, and all its derivatives bounded. Besides, the construction is generalized to the interpolation of the values of the function and its derivatives up to an arbitrarily prescribed order ( Hermitian interpolation). The proposed interpolant depends on a free parameter λ, and its behavior as λ varies is illustrated by a numerical example.