It is often stated that a single-valued analytic function f(z) is uniquely and completely determined if (1) it is analytic at all points of a convergent sequence of points in the complex plane and at their limit point and (2) one is given the points of the sequence and the values of f(z) at each of these points.
Let z(1),z(2),...,z(n)... be a convergent sequence of complex numbers which are strictly decreasing in absolute value as n increases, and whose limit point is zero. Let f(z) be analytic at all the points of this sequence and at their limit point. Supposing that for each positive integer i one is given z(i) and f(z(i)). Does there then always exist a unique power series P(z) centered at zero such that (3) the radius of convergence R of P(z) is positive (or infinite) and (4) if k is any positive integer for which the absolute value of z(k) is less than R, P(z(k))=f(z(k))?
If such a unique power series exists, how do we obtain its coefficients from the data we are given? One can set up an equation for these unknown coefficients involving two infinite column vectors and an infinite Vandermonde matrix. The rows of the matrix are all of the form 1,z(j),z(j)^2,z(j)^3...where j is a positive integer. But I do not know what conditions are needed to insure that such matrices have a unique inverse.