Real Analysis – How to Perform Analytic Lagrange Interpolation for Countably Infinite Points

Question

Suppose I have a finite set of points on the real plane, and I want to find the univariate polynomial interpolating all of them. Lagrange interpolation gives me the least-degree polynomial going through all of those.

Is there an analogous construct for a countably infinite, sparse set of points on the real plane, instead using analytic functions and power series?

There is obviously some difficulty in forming a perfect analogy, as Lagrange interpolation yields the "lowest degree" polynomial interpolating the points, whereas there is no such thing as a "lowest degree" power series. However, perhaps there is some generalized measure of the complexity of a power series that is decently workable, and which restricts to the lowest-degree polynomial in the finite case.

If so, how does this work? Is there an easy way to obtain the nth coefficient of the power series from the points?

Answer 1

There is this theorem:

Given two sequences $z_n$ and $w_n$ of complex numbers such that $|z_n| \to \infty$, there exists a holomorphic function $f$ such that $f(z_n) = w_n$ for all $n$.

It is a consequence of the Weierstrass factorization theorem and the Mittag-Leffler theorem.

See this question.