06/12/2022, 09:56 PM
Interpolating is actually pretty easy. It's when you ask for a functional equation that it's difficult.
Assume \(a_n \to \infty\) and \(b_n\to\infty\) and we want to find \(f(b_n) = a_n\). Define a Weierstrass function \(W(z)\), such that \(W(b_n) = 0\). Then define:
\[
f(z) = W(z) \sum_{n=0}^\infty \frac{a_n}{W'(b_n)(z-b_n)}\\
\]
You can choose \(W\) such that the series converges, and that's pretty much it.
This is an exercise in John B Conway's complex analysis, if you're looking for a source.
Assume \(a_n \to \infty\) and \(b_n\to\infty\) and we want to find \(f(b_n) = a_n\). Define a Weierstrass function \(W(z)\), such that \(W(b_n) = 0\). Then define:
\[
f(z) = W(z) \sum_{n=0}^\infty \frac{a_n}{W'(b_n)(z-b_n)}\\
\]
You can choose \(W\) such that the series converges, and that's pretty much it.
This is an exercise in John B Conway's complex analysis, if you're looking for a source.