Consider an entire function $\ f:\mathbb C\rightarrow\mathbb C.\ $ Let $\ (a\_n\in\mathbb C:n=0\ 1\ \ldots)\ $ be an infinite sequence, where $\ a\_k\ne a\_n\ $ whenever $\ k\ne n.\ $ Let $\ L\_n\ $ be the the degree $\le n$ polynomial $\ L\_n\ $ such that $\ L\_n(a\_k) = f(a\_k)\ $ for every $\ k=0\ \ldots\ n.\ $ What are the results about convergence of $\ L\_n\ $ to $\ f\ $, e.g. uniform convergence on compact subsets of $\ \mathbb C\,?$



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> The results may depend on the nature of the sequence $\ a\_n\ $ or on a subclass of the entire functions to which $\ f\ $ belongs, or perhaps on both. (Myself, I don't know any such results, be them positive or negative).
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Convergence of Lagrange interpolation polynomials to entire functions
There are too many results to survey them here. The principal books addressing this question are:


B. Levin, Distribution of zeros of of entire functions,


A. Gelfond, Calculus of finite differences,


J. M. Whittaker, Interpolatory function theory.


All these books exist in multiple editions, and can be found on Internet.
But the general form of the results is as you wrote: under certain restrictions
on the nodes and values, Lagrange polynomials (or Lagrange series) converge,
without any restrictions it diverges.


