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.
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 ?