Let $M$ be a nice topological space with a nice measure, let's say a Riemannian manifold. Let $B(0,R)$ be a ball in $\mathbb C$ and suppose $a_n : M \to \mathbb C$ are continuous $L^2$ functions such that the series $$f(w, s) = \sum_{n=0}^\infty a_n(w) s^n$$ converges uniformly on compact subsets of $M \times B(0,R)$. Suppose that each $f(\cdot, s)$ is $L^2$. Suppose also that the $L^2$-norms $\|a_n\|$ grow at most exponentially with $n$.
Does it follow that $\|a_n\| \ll r^{-n}$ for all $r < R$?
Note:
This is immediate if $M$ is compact, because uniform convergence of the series on compact sets then implies $L^2$-convergence. I'm interested in the noncompact case.
This is false when $s$ varies in a real interval. For example, let $$f(w,s) = \frac{ws}{1+(ws)^2}$$ on $\mathbb R \times \mathbb R$ and take the Taylor expansion at any point $s$ that is not $0$.
Broader context. I'm curious whether if a continuous $f$ is holomorphic for fixed $w$ and $L^2$ for fixed $s$, then $f$ is holomorphic as an $L^2$-valued function. I managed to reduce this to the above. (Some of the conditions are automatic, but I added them anyway.)
Is there any literature on this question?