When can you reconstruct the power series of a function by taking the limits of the coefficients of the polynomials that interpolate its values at $0,1,2,\dots,j$?


More precisely:


Let $f\colon\mathbb{R}\to\mathbb{R}$. For all nonnegative integers $j$, let $p\_j$ be the unique polynomial of degree $j$ with real coefficients such that $p\_j(n)=f(n)$ for $n=0,\dots,j$. Let $a\_{j,k}$ be the coefficient of $x^k$ in $p\_j$, where $a\_{j,k}=0$ if $k>j$. Under what circumstances do we have that $\lim\_{j\to\infty} a\_{j,k}$ exists for all $j$? In this case, let $c\_k=\lim\_{j\to\infty} a\_{j,k}$. When $f$ is analytic, under what circumstances do we have that $f(x)=c\_0+c\_1 x+c\_2 x^2 + c\_3 x^3 +\cdots$?



Power series whose coefficients are limits of coefficients of polynomial interpolations
In the paper, A note on convergence of Newton interpolating polynomials,
by D. Dimitrov and J. Philipps, Journal of Computational and Applied Mathematics
Volume 51, Issue 1, 30 May 1994, Pages 127-130, the following simple criterion is mentioned: $f$ is entire of exponential type less than $\log 2$,
then the sequence of interpolation polynomials interpolating at non-negative integers converges to $f$.


