Let $\psi_j$ be a sequence of comlex numbers such that $\sum\limits_{j=0}^{\infty}|\psi_j| < \infty$. Then $$ f(z) = \sum_{j = 0}^{\infty}\psi_jz^j $$ is a holomorphic function on $|z|<1$ since the series converges absolutely. Now let $g(z)$ be a complex polynomial.
I want to understand/justify the following relation:
$$ f(z)g(z) = \sum_{j=0}^{\infty}\varphi_j z^{j} \qquad \text{for} \quad |z|\leq 1. $$
My ideas:
- $f(z)g(z)$ is holomorphic on $|z|<1$ as a product of two holomorphic functions on $|z|<1$. So there exists a Laurent series for the product on the open unit ball $|z|<1$.
- To justify the above representation for the closed ball $|z| \leq 1$, I think of considering a holomorphic extension of $f(z)g(z)$ which would also be defined in the neighborhood of the unit circle and coincide with $f(z)g(z)$ on $|z| \leq 1$.
- Or maybe I should consider a holomorphic extension of $f(z)$.
But I am unsure whether it is possible or whether there is a better way to understand/interpret/justify the above equation.