Question: Hadamard-Weierstrass Factorization Theorem generalization

I have been working on a problem related to and using the famous Hadamard-Weierstrass Factorization Theorem (HWFT) for representing an entire function, E(z), with pre-defined zeroes, a(n), which go off to infinity. From HWFT one can represent any meromorphic function with pre-defined poles and zeroes as the ratio of two entire functions.

I am not interested in creating an entire function, but a function F(z) analytic on a disk centered at a pre-defined point such that the analytic continuation, A(z), of F(z) equals pre-defined values

A(z(n)) = w(n) for an infinite sequence of complex numbers, z(n) and w(n)

and A(z) has singularities: algebraic, essential, logarithmic, poles, at pre-defined points (so all disks on which A(z) is analytic must avoid these points) - possibly countably infinitely many of them

Bascially, I have been trying to generalize the Lagrange Interpolation Formula, but for infinitely many points.  However, having A(z(n))=w(n) for infinitely many z(n), w(n) n=1, 2... where no accumulation points occur in either sequence z(n) nor w(n), does not uniquely define A(z), e.g.

c*sin(z)

for any nonzero complex number, c, has exactly the same infinite set of zeroes, the integer multiples of pi.

I have also been considering the case where z(n) has an accumulation point, e.g. z(n) =1/n but w(n) does not. Would this not uniquely define A(z) at z=0?

What theorems exist that restrict how analytic functions can be represented, and in particular, whether the Lagrange Interpolation Formula - done usually for polynomials, but works for any arbitrarily chosen FINITE collection of functions - works when trying to match/fit an INFINITE number of points ?